If $\phi \colon R \to S$ is a ring homomorphism, one obtains the extension and restriction of scalars functors $\phi_* \colon R\mathbf{Mod} \to S\mathbf{Mod}$ and $\phi^* \colon S\mathbf{Mod} \to R\mathbf{Mod}$ between the module categories of $R$ and $S$.
If $f \colon X \to Y$ is a continuous map of compact Hausdorff spaces, then one gets an induced map $\phi = C(f) \colon C(Y) \to C(X)$ of the rings of complex-valued functions on the spaces given by precomposition by $f$. In this case, a finitely generated projective module $\mathcal{E}$ over $C(Y)$ is mapped via extension of scalars to a finitely generated projective module $\phi_*(\mathcal{E})$ over $C(X)$. Moreover, if $E \to Y$ is a complex vector bundle such that its continuous sections $\Gamma(E)$ is isomorphic to $\mathcal{E}$ as a $C(Y)$-module, then the module $\phi_*(\mathcal{E})$ over $C(X)$ is isomorphic to the continuous sections of the pullback bundle $f^*(E)$ over $X$. In this way, we get a description of the extension of scalars functor along the map $C(f)$ in terms of vector bundles.
Is there a similar characterization of the restriction of scalars functor along $\phi = C(f)$? That is, if $F \to X$ is a vector bundle over $X$, does the $C(Y)$-module $\phi^*(\Gamma(F))$ correspond (at least in some nice cases) to the continuous sections of a vector bundle over $Y$?
EDIT: If $E \to X$ is a vector bundle over $X$, then the only sensible way I can think of to define a "fiber" of a pushforward bundle $f_*E$ at $y \in Y$ would be $(f_*E)_y = \bigoplus_{f(x) = y} E_x$. However, for this to give a (constant rank) vector bundle, one needs the cardinality of the fibers $\{ x \in X: f(x) = y \}$ of $f$ to be constant and finite in $y$, which is of course often not the case.
However, if $X$ and $Y$ are topological groups and and $f$ is a surjective group homomorphism with finite kernel, then the above should actually be satisfied. Are there any results or references that mention this situation?