Let $G$ be a periodic locally soluble group with finite Sylow p-subgroups for all primes p.
It is know that in these conditions $G$ is residually finite. Moreover it can be proved that $G$ has only finitely many conjugacy classes of finite subgroups of given order.
Let $L$ be a subgroup of $G$ such that $L=HN$, where $H$ is a finite subgroup of $G$ and $N$ is a normal subgroup of $G$.
Why the index $|N_G(L):N_G(H)N|$ is finite? I know this is true since this is stated at Lemma 1.6 of the paper "Locally inner endomorphisms of SF-Groups" by Belyaev