Consider a matrix $\bf A$ which satisfies
$$ \bf A = \bf B + (\bf B - \bf I)\bf C (\bf B - \bf I), $$
where $\bf I$ is the identity matrix, $\bf B$ is symmetric and idempotent, and $\bf C$ is symmetric with the modulus of its eigenvalues less than or equal to $1$.
I want to show that all eigenvalues of $\bf A$ have modulus less than or equal to $1$.
This might not be true, but I have not been able to find a counter-example. This fact might be helpful:
$$ \bf A\bf B = \bf B\bf A=\bf B. $$
$B$ is symmetric and idempotent, i.e. it is an orthogonal projection. So, by a change of orthonormal basis, we may assume that $B=0_{k\times k}\oplus I_{n-k}$. Hence (with respect to this basis) $A=C_k\oplus I_{n-k}$, where $C_k$ is the leading principal submatrix of $C$. Now the spectral radius of $C_k$ is bounded above by the spectral radius of $C$, which by assumption is at most $1$. Therefore $\rho(A)\le1$.