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No, because the map on cochains is $f\mapsto \sum_{g\in G/H}gfg^{-1}$ where $f$ is $H$-linear, so that any other coset representative $gh$ would not have any effect.
This shows that $cor^G_H$ is independent of the choice of transversal on the level of cohomology. But in that paper of Eunmi Choi, a $\textit{different}$ map on the level of cochains is given (using the bar resolution), which also induces $cor^G_H$ on cohomology, and this $\textit{is}$ dependent on the choice of transversal, as is shown in that paper. But all such images lie in the same cohomology class.
No, because the map on cochains is $f\mapsto \sum_{g\in G/H}gfg^{-1}$ where $f$ is $H$-linear, so that any other coset representative $gh$ would not have any effect.