Quasirecognizability of simple unitary groups over fields of even order

We refer to the set of element orders of a finite group as the spectrum of this group and say that two groups are isospectral if their spectra coincide. We prove that finite simple unitary groups of dimension at least $5$ over fields of characteristic $2$ other than $U_5(2)$ are quasirecognizable by spectrum, that is every finite group isospectral to such unitary group $U$ has a unique nonabelian composition factor and this factor is isomorphic to $U$.