Let $ G $ is soluble group with $ \Phi(G) = 1 $ and assume that each minimal normal subgroup has prime order or order $ 4 $. Let $ K/L $ be a chief factor of $ G $ and $ M $ be the smallest normal subgroup of $ K $ such that $ K/M $ is nilpotent, and so $ M \lhd G $ and $ M \leq L $. If $ M = 1 $, then $ K $ is nilpotent, so $ K \leq F(G) $. Then why $ K/L $ is $ G $-isomorphic to a normal minimal subgroup of $ G $ of order $ 4 $ ?
Two chief factors of $G$ are $G$-isomorphic if there is an isomorphism $f$ between them such that $f(n^g)=f(n)^g$ for all $g \in G$.