I am wont to give the following generalization of this problem
for homework/exams:
Two plane conducting plates of equal area are placed parallel
to each other in air. A charge Q1 is placed on one plate;
a charge Q2 is placed on the other. Using the usual "infinite
sheet" approximation, determine the amount of charge on each
of the four surfaces, in the electrostatic steady state.
(Each charge can be positive or negative.)
The answers are:
Q1_outside = 0.5(Q1 + Q2)
Q1_inside = 0.5(Q1 - Q2)
Q2_inside = 0.5(Q2 - Q1))
Q2_outside = 0.5(Q1 + Q2)
Interesting special cases are Q1=Q2; Q1=-Q2; Q1=2*Q2; Q2=-2*Q2
The analysis consists of imposing conservation of total charge
and E=0 inside a conductor (as John has pointed out).