Is there a name for the operation $f^{-1}(f(x) \oplus f(y))$?

These are all instances of conjugation. The operation $(x,y)\mapsto f^{-1}(f(x)\oplus f(y))$ is a conjugate of the operation $(u,v) \mapsto u \oplus v$.


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.

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