Groups isospectral to the degree 10 alternating group

The spectrum of a finite group is the set of its element orders. We describe the composition structure of every finite group with the same spectrum as that of the alternating group of degree 10 and not isomorphic to it. This group is isomorphic to the semidirect product of the abelian $\{3,7\}$-group, which contains an element of order 21, by the symmetric group of degree 5.