Suppose I create a persons face in Inkscape using separate images, which I also created in Inkscape such as eyes, mouth, nose and ears. And then I group that image into one single image of a face. Is it possible to perform a path operation on that image? For example if I decided to cut the image of the face in half vertically, is that possible using a path operation? I tried super imposing a rectangle over half of the face and then doing the Path Difference operation, however that did not work? So if anyone knows how I can cut an image in half please let me know. Thank you.

Difference doesn't work because one object is the group and status bar says : "One object is not a path : cannot perform bolean operations".

Depending of your goal :

- you can use a clip instead of path>difference : the result is non destructive and editable.

1. first select the group (let's name it the "first group") that regroup all the shapes of the face. Group it again (on status bar, you then have "Group of 1 object in layer...") : let's name this one the "new group" 
2. select "new group" and the rectangle you did design to do path>difference a do Object>Clip>Set.
3. entering in the new group will after that allow you to move the first group. You can also enter the first group to move one ear, the lips...). You can also remove the clip if your not happy with the result. You might also try to use a mask instead of a clip for a progressive transition.

- you can install multiple bolean operations extension to do what you wanted to do : this is a destructive operation : work with copies or ctrl-z

https://inkscape.org/~Moini/%E2%98%85multiple-boolean-operations-with-inx-pathops

Not sure it works with groups : you might have to ungroup the face and select all its parts  and the rectangle.

Multiple boleans operations can work with group but you need to use the PathOps custom and check "Recurse into groups"  option.

See attached file : wether there are subgroups or not, you will get issues with Difference if you don't use this option.