In Selick's very pretty paper "Odd primary torsion in $\pi_*(S^3)$" he makes use of an automorphism which was established by Toda in his paper "On the double suspension $E^2$".
Unfortunately, Selick just refers to [10], but the paper is about 40 pages and dense reading; after skimming it several times, I fear that the automorphism is there either implicitly, or with different notation.
Can someone provide a more precise reference? A page or theorem number would be ideal.
Do you mean the argument in the last page of Selick's paper?
If so, he does not make use of an "automorphism". What he wanted to say is the difference between the map $$(H'\circ\Omega\gamma)_* : H_*(\Omega S^{2p+1}\{p\}) \longrightarrow H_*(\Omega S^{2p+1}_{(p)})$$ and the map $(\Omega i)_*$ is given by a nonzero multiple of an element of $\mathbb{F}_p$ on the generator $y_{2p}$.
You don't need to read Toda's calculations. Generators of these homology Hopf algebras are given in Selick's paper. All you need to do is the image of $y_{2p}$ is nontrivial. This can be done by using the behavior of the mod $p$ Hopf invariant $H : \Omega S^3 \to \Omega S^{2p+1}$ on homology.
By definition, $H$ is designed to be an isomorphism on homology in degree $2p$.
