Injections into function spaces over compacta

We study the topology of X given that Cp(X) injects into Cp(Y), where Y is compact. We first show that if Cp over a GO-space injects into Cp over a compactum, then the Dedekind remainder of the GO-space is hereditarily paracompact. Also, for each ordinal τ of uncountable cofinality, we construct a continuous bijection of Cp(τ,{0,1}) onto a subgroup of Cp(τ+1,{0,1}), which is in addition a group isomorphism.