On $$mathfrak {F}_{hq}$$-supplemented subgroups of a finite group

A subgroup H of a finite group G is quasinormal in G if it permutes with every subgroup of G. A subgroup H of a finite group G is (mathfrak {F}_{hq})-supplemented in G if G has a quasinormal subgroup N such that HN is a Hall subgroup of G and ((Hcap N)H_{G}/ H_{G} le Z_{mathfrak {F}}(G/H_{G})), where (H_{G}) is the core of H in G and ({Z}_{mathfrak {F}} (G/H_{G})) is the (mathfrak {F})-hypercenter of ({G/H}_{G}). This paper concerns the structure of a finite group G under the assumption that some subgroups of G are (mathfrak {F}_{hq})-supplemented in G.