Arithmetic properties of the 2-color overpartition function ${\bar{p}}_{\mathrm{\ell }}(n)$

We derive general families of Ramanujan-type congruences for the function ${\bar{p}}_{\mathrm{\ell }}(n)$, which counts the $2$-color overpartitions of $n$ in which one of the colors appears only in parts that are multiples of $\mathrm{\ell }$. For example, for all $n\ge 0$ we prove that $${\bar{p}}_4(32n+28)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}512)\mathrm{.}$$