The error is that Rokhlin claimed that $\pi_6(S^3)=\mathbb{Z}/6$, but Hilton, in his review, points out that the paper instead shows that $\pi_6(S^3)/\pi_5(S^2) = \mathbb{Z}/6$. The error lies in a prior calculation (reviewed here) that Rokhlin claimed showed $\eta^3=0$, but in fact this element is 2-torsion.

Rokhlin corrects his mistake and calculates the stable homotopy group $\pi_3^s$ in

Rohlin, V. A. MR0052101
New results in the theory of four-dimensional manifolds. (Russian)
Doklady Akad. Nauk SSSR (N.S.) 84, (1952). 221–224.

The review states that this result "agrees with, and were anticipated by, results of Massey, G. W. Whitehead, Barratt, Paechter and Serre." Serre's CR note Sur les groupes d'Eilenberg-MacLane. C. R. Acad. Sci. Paris 234, (1952). 1243–1245. found the correct $\pi_6(S^3)$ by homotopical means. Barratt and Paechter found an element of order 4 in $\pi_{3+k}(S^k)$ when $k\geq 2$.

The reference to Massey-Whitehead you no doubt saw is a result presented at the 1951 Summer Meeting of the AMS at Minneapolis; all we have is the abstract in the Bulletin of the AMS 57, no. 6

If one wants to analyse 'dates received' to establish priority, then by all means.

The error is that Rokhlin claimed that $\pi_6(S^3)=\mathbb{Z}/6$, but Hilton, in his review, points out that the paper instead shows that $\pi_6(S^3)/\pi_5(S^2) = \mathbb{Z}/6$. The error lies in a prior calculation (reviewed here) that Rokhlin claimed showed $\eta^3=0$, but in fact this element is 2-torsion.

Rokhlin corrects his mistake and calculates the stable homotopy group $\pi_3^s$ in

Rohlin, V. A. MR0052101
New results in the theory of four-dimensional manifolds. (Russian)
Doklady Akad. Nauk SSSR (N.S.) 84, (1952). 221–224.

The review states that this result "agrees with, and were anticipated by, results of Massey, G. W. Whitehead, Barratt, Paechter and Serre." Serre's CR note Sur les groupes d'Eilenberg-MacLane. C. R. Acad. Sci. Paris 234, (1952). 1243–1245 (BnF) found the correct $\pi_6(S^3)$ by homotopical means. Barratt and Paechter found an element of order 4 in $\pi_{3+k}(S^k)$ when $k\geq 2$.

The reference to Massey-Whitehead is a result presented at the 1951 Summer Meeting of the AMS at Minneapolis; all we have is the abstract in the Bulletin of the AMS 57, no. 6

If one wants to analyse 'dates received' to establish priority, then by all means.