Let $G$ be a group with $N$ a normal subgroup (not necessarily of finite index). Let $Q$ and $A$ be $G$-modules and $P$ be a $G/N$-module.
I want to make sense of the term ${\mathrm{Hom}}_{G/N}(P, {\mathrm{Hom}}_N(Q,A))$.
Here $Q$ and $A$ are $N$-modules by restriction of the $G$-action. Is there any standard $G/N$-module structure on ${\mathrm{Hom}}_N(Q,A)$?
${\mathrm{\bf{Some~background~on~the~question:}}}$ If $N \unlhd G$, and $A$ is a $G$-module, then there is a (Hochschild-Serre-) spectral sequence $H^p(G/N,H^q(N,A)) {\Rightarrow}_p H^{p+q}(G,A)$.
The construction of this begin with taking a pair of projective resolution ${\mathrm{\bf{P}}} \rightarrow {\mathbb{Z}}$ (as $G/N$-modules) and ${\mathrm{\bf{Q}}} \rightarrow {\mathbb{Z}}$ (as $G$-modules) and form the double complex $E^{0}_{p,q} := {\mathrm{Hom}}_{G/N}(P_p, {\mathrm{Hom}}_N(Q_q,A))$ ($0$-th level).
Let $f\colon Q \to A$ be an $N$-module homomorphism, let $g \in G$, and let $q \in Q$. Then we can define $$(g * f)(q) = g f(g^{-1} q).$$ If $g \in N$, then $g * f = f$ because $f$ is $N$-equivariant. It's a straightforward check of the definitions to verify that $g * f$ defined as above is an $N$-equivariant linear map and that this defines a group action of $G/N$ on $\operatorname{Hom}_N(Q, A)$. (It might be helpful to think of $G/N$ as "acting by conjugation" here.)