Given an open subscheme $f: U\hookrightarrow X$ and an injective etale sheaf of abelian groups $\mathcal{I}$ on $U$, then is it necessarily true that $f_!(\mathcal{I})$ is also injective?
If not then how one proves that $H^i(X, f_!(\mathcal{F})) = H_c^i(U, \mathcal{F})$ when $X$ is proper?
Although I still don't know whether $f_!$ preserves injectivity or not (it should be incorrect in general but might be true for open immersions) but the last statement follows from proper base change.
