Let $S$ be a noetherian scheme (possibly nicer assumptions) and $G$ a smooth group scheme of finite type over $S$ with geometrically integral fibers with structure morphism $f: G\rightarrow S$ and $f_*\mathcal O_G =\mathcal O_S$ universally.
For $n\geq 0$ let $G_n$ denote the $n$-th infinitesimal neighborhood of the zero section of $S$ in $G$ and by denote $f_n: G_n \rightarrow S$ its structure morphism and denote by $g_n: S \rightarrow G_n$ the obvious closed immersion.
Let $\mathcal V$ be a coherent sheaf on $G_n$ with the following property:
$(*)$ The direct image of $\mathcal V$ along $f_n$ is locally free of finite type on $S$.
The question is if one can conclude that then the following holds:
$(**)$ The pullback $g_n^* \mathcal V$ of $\mathcal V$ along $g_n$ is locally free of finite type on $S$.
One may feel free to add additional assumptions on $G$ or $S$ or $\mathcal V$ in a discussion about when $(**)$ would hold, but the general case is of course very interesting.
Note: One could of course also ask if $(*)$ already implies that $\mathcal V$ itself is locally free of finite rank, but I guess this would be false in general.