Jacobian varieties with group algebra decomposition not affordable by Prym varieties

The action of a finite group $G$ on a compact Riemann surface $X$ naturally induces another action of $G$ on its Jacobian variety $\mathrm{J}(X)$. In many cases, each component of the group algebra decomposition of $\mathrm{J}(X)$ is isogenous to a Prym varieties of an intermediate covering of the Galois covering ${\pi }_G\text{\hspace{0.17em}⁣}:X\to X\mathrm{/}G$; in such a case, we say that the group algebra decomposition is affordable by Prym varieties. In this article, we present an infinite family of groups that act on Riemann surfaces in a manner that the group algebra decomposition of $\mathrm{J}(X)$ is not affordable by Prym varieties; namely, affine groups $\mathrm{Aff}({\mathbb{F}}_q)$ with some exceptions: $q=2$, $q=9$, $q$ a Fermat prime, $q=2^n$ with $2^n-1$ a Mersenne prime and some particular cases when $X\mathrm{/}G$ has genus $0$ or $1$. In each one of this exceptional cases, we give the group algebra decomposition of $\mathrm{J}(X)$ by Prym varieties.