Is there a name for the operation $f^{-1}(f(x) \oplus f(y))$?
Solution 1:
These are all instances of conjugation. The operation $(x,y)\mapsto f^{-1}(f(x)\oplus f(y))$ is a conjugate of the operation $(u,v) \mapsto u \oplus v$.
Solution 2:
There are probably several names around, but for me it is called "the operation obtained by transport of structure". A nice example is the one I wrote down in this answer.