Rank stability of elliptic curves in certain non-abelian extensions

Let $E_{\mathrm{/}\mathbb{Q}}$ be an elliptic curve with rank $E(\mathbb{Q})=0$. Fix an odd prime $p$, a positive integer $n$ and a finite abelian extension $K\mathrm{/}\mathbb{Q}$ with rank $E(K)=0$. In this paper, we show that there exist infinitely many extensions $L\mathrm{/}K$ such that $L\mathrm{/}\mathbb{Q}$ is Galois with $\mathrm{Gal}(L\mathrm{/}\mathbb{Q})\simeq \mathrm{Gal}(K\mathrm{/}\mathbb{Q})⋉\mathbb{Z}\mathrm{/}{p}^n\mathbb{Z}$, and rank $E(L)=0$. This is an extension of earlier results on rank stability of elliptic curves in cyclic extensions of prime power order to a non-abelian setting. We also obtain an asymptotic lower bound for the number of such extensions, ordered by their absolute discriminant.