1 . Let $f: (X,\mathcal O_X) \rightarrow (Y,\mathcal O_Y)$ be a morphism of ringed spaces, and let $\mathcal G$ be a sheaf of $\mathcal O_Y$-modules. Define the inverse image $f^{\ast} \mathcal G$ to be the sheaf of $\mathcal O_X$-modules
$$f^{-1} \mathcal G \otimes_{f^{-1}\mathcal O_Y} \mathcal O_X$$
where $f^{-1}\mathcal G$ is the inverse image sheaf.
2 . Let $\phi: G \rightarrow G'$ be a continuous homomorphism of topological groups of td type, and let $H(G)$ be the Hecke algebra of $G$ (the space of locally constant compactly supported functions $G \rightarrow \mathbb{C}$, with convolution as multiplication). Define a right $H(G)$-module structure on $H(G')$ by setting
$$(f' \ast f)(g') = \int\limits_{G} f'(g'\phi(g))f(g) \delta_{G'}(\phi(g)^{-1}) dg $$
where $\delta_{G'}$ is the left modular character of $G'$. Then to any nongenerate $H(G)$-module $V$ (that is, any smooth representation of $G$), define $f_{\ast}(V)$ to be the $H(G')$-module
$$H(G') \otimes_{H(G)} V$$
The reference for this is Representations of $\mathfrak p$-adic groups by Pierre Cartier, Corvallis proceedings.
There is clearly something in common between the functors defined in (1) and (2), but I do not know what it could possibly be. Is there a nice categorical way of thinking about these two notions?