Intersection of finite-dominating subgroup with any subgroup whose product with it is the whole group is finite-dominating in it

Statement

Suppose $G$ is a group, $H$ is a Finite-dominating subgroup (?) of $G$, and $K$ is any subgroup of $G$ such that $HK=G$. Then, $H\cap K$ is a finite-dominating subgroup of $K$.

This statement can be interpreted as saying that the property of being a finite-dominating subgroup satisfies a weak variant of the transfer condition.

Proof

Given: $H\leq G$ is a finite-dominating subgroup, $HK=G$.

To prove: $K\cap H$ is finite-dominating in $H$. In other words, if $F$ is a finite subgroup of $K$, $F$ is conjugate in $K$ to a subgroup of $K\cap H$.

Proof: Since $H$ is finite-dominating in $G$, there exists $g\in G$ such that $gFg^{-1}\leq H$. Since $G=HK$, we can write $g=hk$, with $h\in H,k\in K$. Then, $hkFk^{-1}h^{-1}\leq H$. Since $h\in H$, this yields $kFk^{-1}\leq H$.

By assumption, $k\in K$, so $kFk^{-1}\leq H\cap K$. Thus, we have found $k\in K$ conjugating $F$ to a subgroup of $H\cap K$.