How does $\operatorname{Ind}^G_H$ behave with respect to $\bigoplus$? How does $\operatorname{Ind}^G_H$ behave with respect to $\bigoplus$? What are the properties of $\operatorname{Ind}^G_H$ with regard to taking direct sums? Any answers or texts to read would be most helpful.

Induced representation is a left adjoint functor, so it preserves colimits, in particular arbitrary direct sums.