Throughout this paper, all groups are finite and G always denotes a finite group. Let σ = { σ i | i ∈ I } be a partition of the set of all primes P . The group G is said to be: σ-primary if G is a σ i -group for some i = i ( G ) ; σ-nilpotent if G = G 1 × … × G n for some σ -primary groups G 1 , … , G n ; σ-soluble if every chief factor of G is σ -primary; σ-full if G possesses a Hall σ i -subgroup for all i such that σ i ∩ π ( G ) ≠ ∅ . A subgroup A of G is said to be σ-permutable in G provided G is σ -full and A permutes with every Hall σ i -subgroup H of G , that is, A H = H A for all i ; G is said to be a PσT-group if σ -permutability is a transitive relation in G , that is, if K is a σ -permutable subgroup of H and H is a σ -permutable subgroup of G , then K is a σ -permutable subgroup of G . Let F be a class of group. Then a set Σ of subgroups of G is called a G-covering subgroup system for the class F if G ∈ F whenever Σ ⊆ F . We prove that: If a set of subgroups Σ of G contains at least one supplement to each maximal subgroup of every Sylow subgroup of G, then Σ is a G-covering subgroup system for the classes of all σ-soluble groups, all σ-nilpotent groups, and all σ-soluble PσT-groups. This result gives positive answers to Questions 19.87 and 19.88 in the Kourovka Notebook and, also, allows us to obtain the following characterization of σ -soluble PσT -groups: G is a σ-soluble PσT-group if and only if each maximal subgroup of every Sylow subgroup of G has a supplement T in G such that T is a σ-soluble PσT-group.