Why $ K \cap H $ is a maximal subgroup of $H $?

Question

Suppose $ G $ is a finite group and $ H \unlhd G $. Suppose that $ H \cap M $ is either $ H $ or a maximal subgroup of $ H $ for any maximal subgroup $ M $ of $ G $. Let $ N $ be a minimal normal subgroup of $ G $. Suppose $ M/N $ is a maximal subgroup of $ G/N $. Then $ M/N \cap HN/N = ( M \cap H )N/N $ equal $ HN/N $ or a maximal subgroup of $ HN/N $. Suppose $ K/N $ is a maximal subgroup of $ HN/N $, then $ K/N = K/N \cap HN/N = (K \cap H)N/N $. Why $ K \cap H $ is a maximal subgroup of $H $?