Assume that $ G = MC $, for some cyclic subgroup $ C $. Is $ M \cap C $ a normal subgroup of $ G $?

Question

Let $ G $ is a solvable finite group and $ M $ be a maximal subgroup of $ G $, and assume that $ G = MC $, for some cyclic subgroup $ C $. If $ M_{G} = 1 $ that $ M_{G} $ is core of $ M $ in $ G $, is$ M \cap C $ a normal subgroup of $ G $?

Answer 1

No: Consider the solvable group $G:=\operatorname{Aff}(\Bbb{F}_5)\cong\Bbb{F}_5\rtimes\Bbb{F}_5^{\times}$ with the maximal subgroup $$M:=\Bbb{F}_5\rtimes\Bbb{F}_5^{\times2},$$ of affine transformations with square leading coefficient, and cyclic subgroup $$C:=\{0\}\rtimes\Bbb{F}_5^{\times},$$ of linear transformations. Clearly $MC=G$ but we have $$M\cap C=\{0\}\rtimes\Bbb{F}_5^{\times2},$$ which is not normal in $G$.