 Solution

Let $G$ act by left multiplication on the coset set $G/H$. Composing the resulting homomorphismwith the sign homomorphism gives a homomorphism $G\to \{\pm 1\}$. If it were nontrivial, its kernel would have index two, contrary to the hypothesis. Thus every element of $G$, and in particular $x$, induces an even permutation.

Since $P$ is a Sylow $2$-subgroup, $[G:P]$ is odd. ThereforeAn involution acts as disjoint transpositions and fixed points. If $x$ fixed no coset, it would be a product of $[G:H]/2$ transpositions, an odd number, contradicting evenness. HenceThis is the even-involution coset fixed-point lemma.