Induced electric field outside a solenoid

I am currently working through an example out of the Barron's AP Physics C book. This example uses Faraday's law to calculate the induced electric field outside a solenoid. The exact problem is:

The current passing through an infinitely long solenoid of radius $$R$$ with $$n$$ turns per length is given as a function of time, $$I(t)$$. What is the magnitude of the electric field at a distance $$a>R$$ from the center of the solenoid?

This is the part of the provided solution that confuses me:

Because of symmetry, the electric field must have a constant magnitude and relative direction throughout the circle. Two possible ways this might be achieved are shown in Figure 18.7. However, even though a radial component of the electric field is allowed to be nonzero by symmetry, Gauss's law requires that it be zero. Imagine drawing a cylindrical gaussian surface with radius $$a$$: Because the gaussian surface encloses no net charge, it must experience zero electric flux. This requires the radial component of the field to be zero. therefore, the electric field must be tangent to the direction of integration.

I understand that if there is no net charge enclosed by the gaussian surface, then the radial component of the electric field must be zero, but it seems to me that there is a net charge within the gaussian surface, else there would be no current through the solenoid. What is wrong with my logic?