Framed instanton homology and Frøyshov's invariant

We determine the framed instanton homology with coefficients in $\mathbb{F}=\mathbb{Z}\mathrm{/}2$ for Dehn surgeries on a knot in the $3$-sphere. The dimension of these groups is seen to have a close relationship with a homology cobordism invariant due to Froyshov. As an application, we show that $r$-surgery on a non-trivial knot cannot be nondegenerate $SU(2)$-abelian for any $\mathrm{\mid }r\mathrm{\mid }\le 4\lceil g(K)\mathrm{/}2\rceil$, which is $2g(K)$ for $g$ even and $2g(K)+2$ for $g$ odd.