Higher homotopy groups

Question

Theorem 5.1 of this paper describes a map $K_n(R)\to \pi_{n+1}(SK(E(R),1))$, where $S$ denotes the suspension. My question: Do we have a map from $K_n(R)\to \pi_{n+1}(S^2K(E(R),1))$. Any reference is appreciated. Basically I would be satisfied if there is a long exact sequence involving $K$ groups and homotopy groups of (iterated)suspensions of $K(G,1)$.