The diameter of the generating graph of a finite soluble group

Let G be a finite 2-generated soluble group and suppose that 〈a1,b1〉=〈a2,b2〉=G. Then there exist c1,c2 such that 〈a1,c1〉=〈c1,c2〉=〈c2,a2〉=G. Equivalently, the subgraph Δ(G) of the generating graph of a 2-generated finite soluble group G obtained by removing the isolated vertices has diameter at most 3. We construct a 2-generated group G of order 210⋅32 for which this bound is sharp. However a stronger result holds if G′ has odd order or G′ is nilpotent: in this case there exists b∈G with 〈a1,b〉=〈a2,b〉=G.