The presence of a charge $Q$ in the cavity produces an electric field that penetrates the walls of the cavity, inducing a charge $Q'$. Since the ball is made of metal, the sphere is conducting, so charge on it resides on its surface, then the electric field inside of it is zero. Applying Gauss law, for a Gaussian sphere centered at the position of charge $Q$ and radius $r$, always inside the ball, $\int \vec{E}\cdot d\vec{S} = \frac{Q+Q'}{\varepsilon_0}$ as $\vec{E} = \vec{0}$, $Q' = -Q$ and the outer surface of ball is charged with $-Q' = +Q$.
Let's take a Gaussian sphere centered at the ball's center with radius $R$, and consider the differential form of Gauss law, $\vec{E}\cdot d\vec{S} = \frac{dq}{\varepsilon_0}$ $\sigma = \frac{dq}{dS} = \varepsilon_0 E(R)$ but E(R) has a constant value independently on cavity position inside the ball. So, $\sigma$ is constant (the distribution over the outer surface is uniform).
Applying Gauss law for a Gaussian sphere of radius $R$ centered at the ball (total charge = Q'-Q' = 0), $\int\vec{E}\cdot d\vec{S} = \frac{-Q'+Q'+Q}{\varepsilon_0} = \frac{Q}{\varepslion_0}$(1) $\sigma = \frac{dq}{dS} = \varepsilon_0 E(R)$(2) From (1), $E(R) = \frac{Q}{4\pi\varepsilon_0 R^2}$(3) Putting (3) into (2), $\sigma = \frac{Q}{4\pi R}$
Applying Gauss law for a sphere centered at ball and radius $r > R$, $\int \vec{E}\cdot d\vec{S} = \frac{q+Q}{\varepsilon_0}$ $E(L) = \frac{q+Q}{4\pi\varepsilon_0 L^2}$ The value of E(L) doesn't depend on the cavity's position nor size of sphere, but the enclosed charge.
