Groups of permutations that are even on maximal proper subsets, and related monoids

Let $n$ be a positive integer and let $[n]=\{1,2,\dots ,n\}$. Let ${\mathrm{\Gamma }}_n$ denote the group of permutations on $[n]$ whose restrictions to maximal proper subsets of $[n]$ are even, let ${\mathrm{\Sigma }}_n$ denote the monoid of transformations on $[n]$ whose injective restrictions to maximal proper subsets of $[n]$ are even and let ${\mathrm{\Delta }}_n$ denote the submonoid of ${\mathrm{\Sigma }}_n$ generated by transformations of rank at least $n-1$. In this paper, we present descriptions of ${\mathrm{\Gamma }}_n$, ${\mathrm{\Delta }}_n$ and ${\mathrm{\Sigma }}_n$, determine their cardinalities and ranks, and provide minimal generating sets for each of them.