The principal W-algebra of ${\mathfrak{p}\mathfrak{s}\mathfrak{l}}_{2\mathrm{\mid }2}$

We study the structure and representation theory of the principal W-algebra ${\mathsf{W}}_{\mathrm{p}\mathrm{r}}^{\mathsf{k}}$ of ${\mathsf{V}}^{\mathsf{k}}({\mathfrak{p}\mathfrak{s}\mathfrak{l}}_{2\mathrm{\mid }2})$. The defining operator product expansions are computed, as is the Zhu algebra, and these results are used to classify irreducible highest-weight modules. In particular, for $\mathsf{k}=\pm \frac{1}{2}$, ${\mathsf{W}}_{\mathrm{p}\mathrm{r}}^{\mathsf{k}}$ is not simple and the corresponding simple quotient is the symplectic fermion vertex algebra. We use this fact, along with inverse hamiltonian reduction, to study relaxed highest-weight and logarithmic modules for the small $N=4$ superconformal algebra at central charges $-9$ and $-3$.