The generalized Van Vleck's equation on locally compact groups

We determine the continuous solutions ʄ, g :G → C of each of the two functional equations ∫G{ʄ,(xyt) – ʄ(σ(y)xt)}dμ(t) = ʄ(x)g(y), x, y ∈ G, ∫G{ʄ,(xyt) – ʄ(σ(y)xt)}dμ(t) = g(x)ʄ(y), x, y ∈ G, where G is a locally compact group, σ is a continuous involutive automorphism on G, and μ is a compactly supported, complex-valued Borel measure on G.