Who computed the third stable homotopy group?

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.

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

This and other matters are discussed in the monograph

[À la recherche de la topologie perdue. (French) [Remembrance of topology past] I. Du côté de chez Rohlin. II. Le côté de Casson. [I. Rokhlin's way. II. Casson's way] Edited by Lucien Guillou and Alexis Marin. Progress in Mathematics, 62. Birkhäuser Boston, Inc., Boston, MA, 1986, MR0900243]

where Rohlin's four papers are reproduced with comments. This book was also translated into Russian (I own a copy) but it seems not into English.