Vtyjkyuk

Let $G$ be a group and $Hleq G$. I am trying to prove that silp$mathbbZHleq$ silp$mathbbZG$ and spli$mathbbZHleq$ spli$mathbbZG$,
where silp$mathbbZG$ (resp. silp$mathbbZH$) is the supremum of the injective dimensions of projective $mathbbZG$-modules (resp. $mathbbZH$-modules) and spli$mathbbZG$ (resp. spli$mathbbZH$) is the supremum of the projective dimensions of injective $mathbbZG$-modules (resp. $mathbbZH$-modules).

That is,
silp$mathbbZG=supid_mathbbZGP$: $P$ is a projective $mathbbZG$-module

and

spli$mathbbZG=suppd_mathbbZGI$: $I$ is an injective $mathbbZG$-module

and silp$mathbbZH$ and spli$mathbbZH$ are defined similarly.

If silp$mathbbZG=infty$, there is nothing to prove. So, let's assume that silp$mathbbZG=n<infty$. If we take a projective $mathbbZG$-module $P$, then we know that $P$ is a projective $mathbbZH$-module, since $mathbbZG$ is a projective $mathbbZH$-module. However, if we start off with a projective $mathbbZH$-module $Q$, I don't see any obvious way to exploit the fact that silp$mathbbZG=n$. We can always take the H-induced $mathbbZG$-module $mathbbZGotimes_mathbbZHQ$. Then $pd_mathbbZGmathbbZGotimesmathbbZHleq n$, but then I don't see how we can infere something about $Q$.

Is there any way to associate every projective (or free) $mathbbZH$-module to a projective (or free) $mathbbZG$-module?

It seems plausible to me since free $mathbbZH$-modules are direct sums of copies of $mathbbZH$, free $mathbbZG$-modules are direct sums of copies of $mathbbZG$ and $mathbbZG$ is itself a direct sum of copies of $mathbbZH$.

However, even if this is true, can something similar be said about injective $mathbbZH$-modules?