If a neutral conductor gets an induced charge and is then grounded, will electrons from the ground flow into the conductors induced positively charged side, resulting in a net negative charge for the whole conductor?
A conductor with a net unbalanced charge, which is then grounded, will acquire whatever charges needed from the ground to become neutral once more.
OTOH: if you induce a separation of charges in a neutral conductor (say, by bringing a charged object close to the conductor) and ground one side of the conductor, then it will be left with an unbalanced charge.
eg. bring a negatively charged sphere close to a neutral conductor to get + charges on the near side and - charges on the opposite. Briefly ground the far side. Remove the sphere. The conductor is now positively charged.
As Simon says: if the charged object which is inducing the separation of charge on the neutral conductor is negatively charged and you briefly ground the neutral conductor on the side furthest away from the charged object, the conductor will then have a net positive charge. If the charged object is positively charged, the conductor will end up with a net negative charge. You'll find ' Lord Kelvin's thunderstorm' to be a fascinating application of electrostatic induction.