Distribution of squarefree integers with double congruence conditions

Given a positive integer $M$ with reduced residue system ${\mathrm{\Phi }}_M$, we partition the set of squarefree integers according to some congruence conditions on the number of prime factors in each class $\phi \in {\mathrm{\Phi }}_M$, and prove two relevant equidsitribution results in this setting. We discover the bias phenomenon in some special cases, and formulate a conjecture with numerical support.