Yep!

Learning algorithms typically find parameters that relate the input to some output.

Genetic algorithms are just a collection of methods that describe how to search through parameter space.

Supervised learning algorithms are ones for which we have some known labels on our inputs (known outputs), whereas in unsupervised learning we don't have any known outputs. Regardless, in both cases we need to learn the parameters that relate the input to the output.

Genetic algorithms can therefore be both supervised or unsupervised.

Does it go both ways(supervised or unsupervised) for the case where I randomly generate my own population and make it evolve into what I anticipate through generations ?

I would think your fitness function is your supervised signal.