Some Comments on Regular Overpartitions modulo $2^k$

For coprime integers $\mathrm{\ell },\mu \ge 2$, Alanazi, Munagi, and Saikia (2026) studied ${\bar{R}}_{\mathrm{\ell },\mu }(n)$, the number of overpartitions of $n$ in which no part is divisible by $\mathrm{\ell }$ or by $\mu$, together with the single-modulus analogue \overline{R}_{\ell (n). We record a simple combinatorial mechanism that determines both functions modulo every power of $2$ in terms of the number of distinct part sizes of the underlying ordinary partition. We also deduce a clean characterization of ${\bar{R}}_{\mathrm{\ell }}(n)$ and ${\bar{R}}_{\mathrm{\ell },\mu }(n)$ modulo $4$ in terms of perfect squares.