= Solution
Let $N$ be a <normal subgroup> and let $\pi:G\to G/N$ be the quotient <group homomorphism>. By the first part, $G=\langle a,b\rangle$ for <involutions> $a,b$, so the <quotient group> is generated by $\pi(a),\pi(b)$, whose squares are the identity. If both images are nonidentity, they are <involutions> and the second part applies. If only one image is nonidentity, it generates $C_2$; choose it twice. Both images cannot be trivial because the <quotient group> is nontrivial. Hence \b[every nontrivial quotient is dihedral in the stated extended convention]. This is the <nontrivial-quotient closure of two-involution groups>.
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