Torsion groups of rational elliptic curves over ${\mathbb{Z}}_p$-extensions of quadratic fields: the $p\le 5$ case

Let $E$ be a rational elliptic curve. We generalize a theorem due to Avcı\cite{AVCI2026153}, which asserts that for any quadratic field $K$ and prime $p>5$, the equality $E(K{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}=E(L{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}$ holds for every ${\mathbb{Z}}_p$-extension $L\mathrm{/}K$. In this paper, we consider the setting where the ${\mathbb{Z}}_p$-extension $L$ is replaced by the compositum $K_{\mathrm{\infty }}$ of all ${\mathbb{Z}}_p$-extensions of $K$. Under this new setting, we prove the analogous statement for $p=5$, and further provide partial results for the remaining primes $p=3$ and $p=2$.