Consider the following statement.

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?