On finite groups isospectral to $PSp_4(q)$

The spectrum of a finite group is the set of its element orders. Let $q$ be a power of a prime $p$, with $p \geqslant 5$. It is known that any finite group having the same spectrum as the simple symplectic group $PSp_4(q)$ either is isomorphic to an almost simple group with socle $PSp_4(q)$ or can be homomorphically mapped onto an almost simple group $H$ with socle $PSL_2(q^2)$. We prove that the group $H$ cannot coincide with $PSL_2(q^2)$, i.e., $H$ must contain outer automorphisms of its socle.