Suppose $X$, $Y$ are finite CW-complexes with free involution and $\mu:X\to Y$ is an equivariant map. If $\mu^*:H^i(Y;\mathbb{Z})\to H^i(X;\mathbb{Z})$ is an isomorphism for $i>i_0$ and is onto for $i=i_0$, then $\mu^{\sharp}:\pi_{\mathrm{eq}}^i(Y)\to\pi_{\mathrm{eq}}^i(X)$ is a 1-1 correspondence for $i>i_0$ and is onto for $i=i_0$.
Moreover, the preimage of each element of $\pi_{\mathrm{eq}}^{i_0}(X)$ is in 1-1 correspondence with the elements of the kernel of $\mu^*:H^{i_0}(Y;\mathbb{Z})\to H^{i_0}(X;\mathbb{Z})$.
Here $\pi_{\mathrm{eq}}^i(Y)$ denotes the set of all equivariant maps $Y\to S^i$ up to equivariant homotopy. This statement, without the 'moreover' part, appears to be known. Becker and Glover in Note on the embedding of manifolds in euclidean space (1971) state it (without the 'moreover' part) and claim that it is well known from obstruction theory. But what about the `moreover' part: is it true? if yes, can you give a reference?