For any $G$-space the $G$-equivariant maps $[EG,X]_G$, also known as the homotopy fixed points $X^{hG}$, are a Borel homotopy invariant of $X$ meaning that it is an invariant of $G$-equivariant maps for which the underlying map is a weak equivalence. If $Z$ has the trivial action, this means we can compute $[EG,Z \times EG]_G$ as $[EG,Z]_G=[EG/G,Z]=[BG,Z].$ On the other hand, the nonequivariant maps $[EG,Z] \cong \pi_0(Z)$ because $EG$ is contractible. So it suffices to find a space, $Z$ such that $[BG,Z] \not= \pi_0(Z)$. The universal example (if $G$ is not contractible) is $Z=BG$.

Explicitly, if $q:EG \rightarrow BG$ is the quotient, this produces homotopic maps $EG \xrightarrow{q \times \mathrm{Id}} BG \times EG$ and $EG \xrightarrow{q \times *} BG \times EG$ which are not equivariantly homotopic.

For any $G$-space the $G$-equivariant maps $[EG,X]_G$, also known as the homotopy fixed points $X^{hG}$, are a Borel homotopy invariant of $X$ meaning that it is an invariant of $G$-equivariant maps for which the underlying map is a weak equivalence. If $Z$ has the trivial action, this means we can compute $[EG,Z \times EG]_G$ as $[EG,Z]_G=[EG/G,Z]=[BG,Z].$ On the other hand, the nonequivariant maps $[EG,Z] \cong \pi_0(Z)$ because $EG$ is contractible. So it suffices to find a space, $Z$ such that $[BG,Z] \not= \pi_0(Z)$. The universal example (if $G$ is not contractible) is $Z=BG$.

Explicitly, if $q:EG \rightarrow BG$ is the quotient, this produces homotopic maps $EG \xrightarrow{q \times \mathrm{Id}} BG \times EG$ and $EG \xrightarrow{* \times \mathrm{Id}} BG \times EG$ which are not equivariantly homotopic.