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Let $p$ be an odd prime and let $k$ be a field of characteristic $p$. Weprovide a practical algebraic description of the representation ring of$k\mathrm{SL}_2(\mathbb{F}_p)$ modulo projectives. We then investigate a familyof modular plethysms of the natural $k\mathrm{SL}_2(\mathbb{F}_p)$-module $E$of the form $\nabla^{\nu}\mathrm{Sym}^l E$ for a partition $\nu$ of size lessthan $p$ and $0\leq l\leq p-2$. Within this family we classify both the modularplethysms of $E$ which are projective and the modular plethysms of $E$ whichhave only one non-projective indecomposable summand which is moreoverirreducible. We generalise these results to similar classifications wheremodular plethysms of $E$ are replaced by $k\mathrm{SL}_2(\mathbb{F}_p)$-modulesof the form $\nabla^{\nu} V$, where $V$ is a non-projective indecomposable$k\mathrm{SL}_2(\mathbb{F}_p)$-module and $|\nu|<p$.

The representation ring of $\mathrm{SL}_2(\mathbb{F}_p)$ and stable modular plethysms of its natural module in characteristic $p$