Is there a way to save selection by faces I'm trying to find a way to save selection faces, tried Vertex Group, the issue with Vertex Group is that it adds the faces that are marked by 4 vertices that are added to the Vertex Group, there are some cases were this doesn't apply, the image explains what I mean. So is there a way to save my selected faces, so I don't have to go through the whole process of reselecting them each time I need to do so?
If you check the image, I didn't assign the hatched faces to the Vertex Group, still when I select the Vertex Group, they get selected, I understand that they got selected because the four boundary vertices are assigned to the Vertex Group, I need to save only face selection, not vertex selection.
This is a limitation of the current Vertex Groups system that has been an occasional nuisance.

That's is why they are called Vertex Groups, not "Face Groups"; selections of vertex don't unequivocally define groups of faces, since a vertex can belong to an arbitrary number of them.

Would be nice to hear about more options and solutions for this from other users around here. The way I generally solve this is by using material slots instead, if you don't mind the additional pollution in the materials list.

Go to Properties Window > Materials Tabs > Add New Material Slot button, and then with the desired faces selected press the Assign button.

Since they work on a per-face basis instead of single vertex, they can correctly define face groups. You can still assign materials independently, but you can't have a face belong to more than one slot unfortunately, so you may end up having several slots with the same materiel.

Different intersections of selections overlapping materials may require an exponentially growing number of slots.