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We present a free field realisation for the vertex operator algebraassociated to the genus-two, class $\mathcal{S}$ superconformal field theory oftype $\mathfrak{a}_1$. The free field realisation is in the style of recentwork by the authors, and is formulated in terms of a one-dimensional isotropiclattice vertex algebra along with two pairs of symplectic fermions. Ourrealisation makes manifest an enhanced ${\rm USp}(4)$ outer automorphism groupof the VOA that is inherited from the symplectic fermion system. This extendsan ${\rm SU(2)}$ outer automorphism that has been observed in recent work ofKiyoshige and Nishinaka and significantly simplifies the structure of thealgebra. Along the way, we also produce a realisation of the generic subregularDrinfel'd-Sokolov $\mathcal{W}$ algebra of type $\mathcal{c}_2$ in terms of thegeneric principle $\mathcal{W}$ algebra of type $\mathfrak{c}_2$ and aone-dimensional isotropic lattice vertex algebra.

A geometric free field realisation for the genus-two class $\mathcal{S}$ theory of type $\mathfrak{a}_1$