It means you have a made profit of $10 on the shares you sold,...

It means you have a made profit of $10 on the shares you sold, on which you pay capital gains tax, and are sitting on a paper profit of $10 for the remaining 5 shares because you bought them for $2 and they are now worth $4.

Your average price on your original investment remains at $2/share. That is different from you total position. Those 5 shares, in theory, have "cost" you nothing if you pay no CGT on the original 5 sold. So, assuming you do pay CGT, not quite free carried as you say. If the sp improves or declines, then that total position changes.

Put it this way, when you sell those 5 remaining shares, what do you want as your cost base for CGT purposes ? $2 is the answer. Not a lower "averaging" position.

Cheers