Let $H_1,H_2\in\mathbb{Q}^{n\times n}$ be idempotent and symmetric matrices. For any $0<\mu<\frac{1}{2}$, consider the matrix

$$H_\mu:=\mu H_1+(1-\mu)H_2.$$

I'm looking for a description of $\text{Eig}(H_\mu,\mu)$.

Clearly, $\text{img}(H_1)\cap\ker(H_2)\subseteq\text{Eig}(H_\mu,\mu)$, but under which conditions do we have equality? Is there a description of the "missing piece"

$$\text{Eig}(H_\mu,\mu)/\text{img}(H_1)\cap\ker(H_2)?$$

See also my question here.