= Solution
Write $H=\{e,h\}$ with $h\ne e$. For any $g\in G$, the <normal subgroup> property implies $ghg^{-1}\in H$. This conjugate cannot be $e$, since multiplication by $g^{-1}$ and $g$ would then give $h=e$. There is only one remaining choice, so
$$
ghg^{-1}=h,\qquad gh=hg.
$$
Thus $h$ commutes with every element of $G$; the identity does too. Hence
$$
\boxed{H\subseteq Z(G).}
$$
This proves that a <normal subgroup of order two is central>. The proof does not require $G$ to be finite: conjugation permutes $H$ and fixes its identity, so it must fix its only other element.
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