  Fix an odd prime number $p$, a finite abelian group $B$ and an elliptic curve
$E/\mathbb{Q}$. Let $K/\mathbb{Q}$ be a Galois extension with
$\text{Gal}(K/\mathbb{Q}) \simeq B$. Suppose that $E(K)$ has rank $0$. Given a
positive integer $n$, let $T \simeq \mathbb{Z}/p^n\mathbb{Z}$ and set
$\mathscr{G} = B \ltimes T$. In this paper, we show that there exist infinitely
many extensions $L/K/\mathbb{Q}$ such that $\text{Gal}(L/\mathbb{Q}) \simeq
\mathscr{G}$ and the rank of $E(L)$ remains $0$. We also prove asymptotic lower
bound for the number of such extensions, ordered by their absolute
discriminant. This extends a result of Mazur and Rubin, who study rank
stability in cyclic extensions of prime power order.
