Groups in which every subgroup has finite index in its Frattini closure

In 1970‎, ‎Menegazzo [Gruppi nei quali ogni sottogruppo \`e intersezione di sottogruppi massimali‎, ‎{\em Atti Accad‎. ‎Naz‎. ‎Lincei Rend‎. ‎Cl‎. ‎Sci‎. ‎Fis‎. ‎Mat‎. ‎Natur.} {\bf 48} (1970)‎, ‎559--562.] gave a complete description of the structure of soluble $IM$-groups‎, ‎i.e.‎, ‎groups in which every subgroup can be obtained as intersection of maximal subgroups‎. ‎A group $G$ is said to have the $FM$-property if every subgroup of $G$ has finite index in the intersection $\hat X$ of all maximal subgroups of $G$ containing $X$‎. ‎The behaviour of (generalized) soluble $FM$-groups is studied in this paper‎. ‎Among other results‎, ‎it is proved that if~$G$ is a (generalized) soluble group for which there exists a positive integer $k$ such that $|\hat X:X|\leq k$ for each subgroup $X$‎, ‎then $G$ is finite-by-$IM$-by-finite‎, ‎i.e.‎, ‎$G$ contains a finite normal subgroup $N$ such that $G/N$ is a finite extension of an $IM$-group‎.