A Polish group containing a Haar null $F_{\sigma }$-subgroup that cannot be enlarged to a Haar null $G_{\delta }$-set

Answering a question of Elekes and Vidny'anszky, we construct a Polish meta-abelian group $H$ and a subgroup $F\subset H$, which is a Haar null $F_{\sigma }$-set in $H$ that cannot be enlarged to a Haar null $G_{\delta }$-set.