The company doesn't want to admit that at first but you are already aware.

This is where negotiation comes into play, and negotiating something higher than what your peers in the company perceive, while remaining in line with reality. Best case is that you move the "average salary" for that position upwards a notch.

Not necessarily. You could certainly be worth less or more to your company than what you're worth on the open market. If it's the former, you probably need a new job.