In a group endomorphism between two sets, must the binary operation of the two sets be the same?

Question

Let $(G,\ast)$ and $(G,\cdot)$ be any two groups, where $\ast$ and $\cdot$ are distinct binary operations. Suppose $\phi:G\to G$ satisfies $$ \phi(x)\cdot \phi(y)=\phi(x\ast y) $$ for all $x,y\in G$. Is $\phi$ a group endomorphism? In other words, for a homomorphism $\phi:G\to H$ to be an endomorphism, do we simply require that the set $G$ equals $H$, or do we also require that the binary operations are equal as functions?

Answer 1

An endomorphism of $(G,*)$ is, by definition, a homomorphism from $(G,*)$ to $(G,*)$. Thus, in your formulation, we require $* = \cdot$ (as functions $G\times G \to G$).

Answer 2

According to https://groupprops.subwiki.org/wiki/Endomorphism_of_a_group,

the criterion is $\sigma(gh)=\sigma(g)\sigma(h)$. Notice they don't refer to two different binary operations.