Nontrivial-quotient closure of two-involution groups

Every nontrivial quotient group of a group generated by two involutions is generated by two involutions, allowing repeated generators. The two generator images have squares equal to the identity. If one image becomes trivial, the other generates $C_2$; if both become trivial, the quotient is trivial and is excluded. Thus the two-involution characterization of a dihedral group is preserved by nontrivial quotients.