Characterize the max and min eigenvalues of a positive definite matrix under congruence transform with semi-definite matrix

Question

I am looking to characterize the maximum or minimum eigenvalues of a matrix $B = A Q A + \mathbf{I}-A$ where

• $A$ is positive semi-definite, symmetric, and has eigenvalues $\lambda_i\in[0,1)$
• $Q$ is positive definite and symmetric
• $\mathbf{I}$ is the identity matrix

It is apparent to me that this is positive definite because it is clearly symmetric and

• $x^\top B x = ||x||^2$ if $x\in Ker(A)$
• $x^\top B x \succeq x^\top A Q A x \succ 0$ if $x\notin Ker(A)$

I am wondering how can I find the maximum and minimum eigenvalues of B with respect to A and Q. If this is not possible, does there exist bounds that satisfy $l \mathbf{I}\preceq B \preceq u\mathbf{I}$ with $l>0$.

Appreciate any help or insight.