The inertia group $I_M$ of a closed $d$-manifold $M$ is the subgroup of $\theta_d$ of h-cobordism classes of homotopy spheres $\Sigma$ such that $\Sigma \# M$ is diffeomorphic to $M$.

Wall and Kosinski proved that $I_{W_g}$ is trivial in all dimensions, where $W_g := \#^g S^n \times S^n$. This resolves your main question: two exotic spheres that are stably diffeomorphic are already diffeomorphic.

I learned this from conversations with Manuel Krannich.

As you explain, it then follows from Kreck's result that the kernel of the map $\eta^B$ agrees with $\theta_{2k}$ itself.

The inertia group $I_M$ of a closed oriented $d$-manifold $M$ is the subgroup of $\theta_d$ of h-cobordism classes of homotopy spheres $\Sigma$ such that $\Sigma \# M$ is diffeomorphic to $M$.

Wall and Kosinski proved that $I_{W_g}$ is trivial in all dimensions, where $W_g := \#^g S^n \times S^n$. This resolves your main question: two exotic spheres that are stably diffeomorphic are already diffeomorphic.

I learned this from conversations with Manuel Krannich.

As you explain, it then follows from Kreck's result that the kernel of the map $\eta^B$ agrees with $\theta_{2k}$ itself.

The inertia group $I_M$ of a closed $d$-manifold $M$ is the subgroup of $\theta_d$ of h-cobordism classes of homotopy spheres $\Sigma$ such that $\Sigma \# M$ is diffeomrophic to $M$.

From Kreck's result outlined in the question, it follows that the kernel of the map $\eta^B$ agrees with $I_{W_g}$, where $W_g := \#^g S^n \times S^n$, if $g$ is sufficiently large.

Wall and Kosinski proved that $I_{W_g}$ is trivial in all dimensions, which resolves your question.

I learned this from conversations with Manuel Krannich.

The inertia group $I_M$ of a closed $d$-manifold $M$ is the subgroup of $\theta_d$ of h-cobordism classes of homotopy spheres $\Sigma$ such that $\Sigma \# M$ is diffeomorphic to $M$.

Wall and Kosinski proved that $I_{W_g}$ is trivial in all dimensions, where $W_g := \#^g S^n \times S^n$. This resolves your main question: two exotic spheres that are stably diffeomorphic are already diffeomorphic.

I learned this from conversations with Manuel Krannich.

As you explain, it then follows from Kreck's result that the kernel of the map $\eta^B$ agrees with $\theta_{2k}$ itself.