On the inverse problem for deformations of finite group representations

Let $s$ be even and $q=p^s$. We show that the ring $W(\mathbb{ F}_{ q})[\![X]\!]/(X^2-pX)$ is a quotient of the universal deformation ring of a representation of a finite group. This amounts to giving an example of a finite group and its $\mathbb{ F}_q$-representation that lifts to $W(\mathbb{ F}_q)$ in two different ways and satisfies certain subtle extra conditions. We achieve this by studying representations of $\mathrm{ SL}(2,\mathbb{ F}_{ p^2})$.

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