Covers and normal covers of finite groups

For a finite noncyclic group G  , let γ(G)γ(G) be the smallest integer k such that G contains k   proper subgroups H1,…,HkH1,…,Hk with the property that every element of G   is contained in Hig for some i∈{1,…,k}i∈{1,…,k} and g∈Gg∈G. We prove that if G is a noncyclic permutation group of degree n  , then γ(G)≤(n+2)/2γ(G)≤(n+2)/2. We then investigate the structure of the groups G   with γ(G)=σ(G)γ(G)=σ(G) (where σ(G)σ(G) is the size of a minimal cover of G  ) and of those with γ(G)=2γ(G)=2.