A conductor can have a zero net charge. One way to do this is to have an inner surface with a net charge of $-Q$ and an outer surface with a net charge of $+Q.$
So for instance you can have a surface charge density of $-Q/(4\pi R^2)$ on the surface $r=R$ and you can have a surface charge density of $+Q/(4\pi (2 R)^2)$ on the surface $r=2R$ and this has zero net charge. You can achieve this by having some electrons leave one surface and leave that surface with a net positive charge and have them arrive at the other surface as extra electrons leaving that surface with a net negative charge. Obviously the total charge stays the same.
Why would this happen? If mobile charges are free to move through the body of the conductor and there were an electric field the charges could flow from one surface to the other and keep doing so until the field was zero inside the body of the conductor.
If the field is zero inside that body of the conductor then if you place a Gaussian surface on a spherical shell of radius $r'$ with $R<r'<2R$ then you get a certain flux. That flux is proportional to the total charge enclosed by the Gaussian surface which in this case includes any charge on the insulator (the entire insulator is enclosed by the Gaussian surface) and any charge on the inner $r=R$ surface of the conductor (the $r=R$ surface is enclosed by the Gaussian surface) and the charge (if any) in the body of the conductor from the inner surface all the way to the Gaussian surface $r=r'.$
If the conductor doesn't have a net charge, then the charge on the outside $r=2R$ surface would have to be the same (equal and opposite) as the charge on the $r=R$ surface (though since they have different areas the surface charge would have to be equal).
So based on the electric field in the body and the charge of the insulator you can find out the charge on the inner surface. Based on the total charge of the conductor and the charge on the inner surface you can find the charge on the outer surface. Then based on the charge on the insulator and the two surfaces you can find the electric field outside the conductor. Each time you use Gauss's law and solve for a field or a charge (which ever you don't know) or you use conservation of charge (the total charge of the conductor is the sum of the charge on the two surfaces).