Let's assume that that's the case.
Can you explain why the character changes in CC before and after I turn on the Active Image Matching Tools?

Is this photo one you can share? We can probably make the description using your own photo to demonstrate.

As Kelley already mentions, you would also have to know the lens settings of the original portrait photo and the angle with respect to the face. Ideally, the original photo would be straight on, but often it is not. For example, I often find that the ears are too high or too low. 

The best Headshot can do is assume a more or less normal face. Headshot photo matching changes the lens setting so that the contour can be lined up with the model, but the only goes so far.

What I mean is that you may try to adjust your model to a face with a somewhat distorted perspective. To get an exact replication under these circumstances would be very hard.

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Remember that iClone isn't set the same way as CC3 is -- Rampa can elaborate (I don't have the time right now).  NO character imported from CC3 into iClone will look the same unless and until you apply some very specific adjustments to the scene.

Let Rampa elucidate (and, again, remember the lens used to take your reference photo plays a BIG part in all of this).

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Alienware Aurora R16, Win 11, i9-149000KF, 3.20GHz CPU, 64GB RAM, RTX 4090 (24GB), Samsung 870 Pro 8TB, Gen3 MVNe M-2 SSD, 4TBx2, 39" Alienware Widescreen Monitor
Mike "ex-genius" Kelley

I don't want any particular "effect". All I want is to make the 3d character look as close to the real person of the photo as possible.
In that case, should I set the Headshot lens value to 32, remorph the character to match the photo, and send it to iClone, where I should set the focal length of the iClone camera at 50mm?
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I just did it. And it doesn't seem to work. 
Now, let's just not talk about the lens for a moment.
The issue I originally have is why the character I make using extensive morph adjustments in CC cannot be imported into iClone as an exact duplicate.

I think you need to experiment.  32mm for a (35mm) camera is a pretty wide angle lens, and one that is NOT flattering to the subject at all.  But it depends on the effect you want.  Do as I said and try a few different views, dollying the camera in or out to get the size right, and see what you like.  

As to what's "accurate" -- a 50mm lens is considered to be the same vision as you eye.  So if you are trying to create a character or make it look as you truly see them, that's the lens you need to shoot for.  It's complicated because the lens that took the original image (the one you are using to match up in CC3) may be a lot wider or narrower than this.  So it's all down to what looks good to you.

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Alienware Aurora R16, Win 11, i9-149000KF, 3.20GHz CPU, 64GB RAM, RTX 4090 (24GB), Samsung 870 Pro 8TB, Gen3 MVNe M-2 SSD, 4TBx2, 39" Alienware Widescreen Monitor
Mike "ex-genius" Kelley

Thanks for the info. I think I see where you're coming from.
The manual linked by Rampa says that the lens value in iClone indicates the focal length of the camera which took the photo. 
The photo file's properties (on Windows 10) show two values:
(1) Focal length = 3mm
(2) 35mm focal length = 32
Since the iClone lens value is between 10 and 70, I think that (2) has to be the value, and that the lens value in iClone should be set at 32. Am I right?
If so, should I set it at 32 and then morph the head again to match the photo?

In addition to what Rampa says, it's also possible the lens in iClone itself isn't matched up.  

Folks look different depending on the lens used by the camera.  iClone has a "digital" equivalent.    If you shoot someone with a telephoto (say a 200mm lens) you will see their features all flatten out (they will look "fatter") but if you shoot someone with a wide-angle lens the features get exaggerated to each other (although the head itself will look thinner.  Ideally a good portrait lens is around 80mm.

This has nothing to do with the relative size of the head, as you can move (dolly) the camera closer or farther to get the same width.  So try it yourself in iClone -- make an 200mm lens and then dolly the camera until the head fills the frame.  Render that image.  Now make a 35mm lens and move the camera closer (dolly it, don't adjust the length of the lens) and compare the two images and you'll see how the face looks very different depending on the lens.

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Alienware Aurora R16, Win 11, i9-149000KF, 3.20GHz CPU, 64GB RAM, RTX 4090 (24GB), Samsung 870 Pro 8TB, Gen3 MVNe M-2 SSD, 4TBx2, 39" Alienware Widescreen Monitor
Mike "ex-genius" Kelley

I don't know what the perspective is, but I don't understand how to adjust the lens. 
Currently, the 3d model is morphed enough to fit the photo. So the 3d model doesn't need any further adjusting. That is, the edges of the 3d model and the photo all line up as is.
How do I go about further adjusting the perspective from here?

I think you may need to adjust the lens a bit. Basically your perspective between the 3D model in the scene, and the photo are mismatched. See "D" on this page of the manual.
https://manual.reallusion.com/Headshot_Plugin/ENU/Headshot_Plugin_for_CC/1.0/ID_SA_IMT_Pro.html
Ajust the perspective so the edges of the head of the 3D character and photo pretty much line up.