A map from a $ G_1 $ - equivariant KK-theory of Kasparov, to a $ G_2 $ - equivariant KK-theory of Kasparov

Question

Let $ G $ be a locally compact group.

Let $ H $ and $ K $ be two normal subgroups of $ G $.

In order to construct a map, $$ \psi \ : \ \ F(G/H,G/K) \to F(G/K,G/H) $$

where, $$ F(G/H,G/K) = KK^{G/H} ( C_0 \big( \underline{E} \big( G/H \big) \big)^{G/K} , C_0 \big( \underline{E} \big( G/K \big) \big)^{G/K} ) $$ i would like to know if, $ C_0 \big( \underline{E} \big( G/H \big) \big)^{G/K} $ and $ C_0 \big( \underline{E} \big( G/K \big) \big)^{G/K} $ are $ G/H $ - $ C^* $ - algebras.

Here,

• $ \underline{E} \big( G/H \big) $ is the universal space for proprer $ G/H $ - actions.

• $ C_0 ( X ) $ is the space of continuous maps, vanishing at infinity.

• $ C_0 ( X )^G = \{ \ f \in C_0 (X) \ | \ f(xg) = f(x) \ , \ \forall (g,x) \in G \times X \ \} $.

I also would like to know how to construct, in general, a map from a $ G_1 $ - equivariant KK-theory of Kasparov, to a $ G_2 $ - equivariant KK-theory of Kasparov, where $ G_1 $ and $ G_2 $ are two distinct groups.

Thanks in advance for your help.