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?