An infinite family of overpartition congruences mod powers of 2

We prove an infinite family of Hecke-like congruences for the overpartition function modulo powers of 2. Starting from a recent identity of Garvan and Morrow and iterating Atkin’s $U_2$ operator, we determine lower bounds on the 2-adic valuations of the coefficients that arise at each step. Our approach yields new modular equations relating the Hauptmoduln $G_2$ on ${\mathrm{\Gamma }}_0(2)$ and $G_8$ on ${\mathrm{\Gamma }}_0(8)$, together with explicit $U_2$-action formulas.