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In this paper, we show that there exist infinitely many extensions $L/K$ such that $L/\\mathbb{Q}$ is Galois with $\\operatorname{Gal}(L/\\mathbb{Q}) \\simeq \\operatorname{Gal}(K/\\mathbb{Q}) \\ltimes \\mathbb{Z}/p^n\\mathbb{Z}$, 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. 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Fix an odd prime $p$, a positive integer $n$ and a finite abelian extension $K/\\mathbb{Q}$ with rank $E(K) = 0$. In this paper, we show that there exist infinitely many extensions $L/K$ such that $L/\\mathbb{Q}$ is Galois with $\\operatorname{Gal}(L/\\mathbb{Q}) \\simeq \\operatorname{Gal}(K/\\mathbb{Q}) \\ltimes \\mathbb{Z}/p^n\\mathbb{Z}$, 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. 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