On finite groups isospectral to PSp4(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 >= 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 PSL2(q(2)). We prove that the group H cannot coincide with PSL2(q(2)), i.e., H must contain outer automorphisms of its socle.