Arithmetic properties modulo powers of $2$ for overpartition $k$-tuples with odd parts

Recently, Drema and Saikia (2023) proved several congruences modulo powers of 2 and 3 for overpartition triples with odd parts. We extend their list substantially. We prove several congruences modulo powers of 2 for overpartition k-tuples with odd parts, along with a few infinite families of congruences for overpartition triples with odd parts and for overpartition k-tuples with odd parts (for k = 4 and for odd k).