On spectra of almost simple groups with symplectic or orthogonal socle

Finite groups are said to be isospectral if they have the same sets of the orders of elements. We investigate almost simple groups $H$ with socle $S$, where $S$ is a finite simple symplectic or orthogonal group over a field of odd characteristic. We prove that if $H$ is isospectral to $S$, then $H/S$ presents a $2$-group. Also we give a criterion for isospectrality of $H$ and $S$ in the case when $S$ is either symplectic or orthogonal of odd dimension.