Transgender and sports

It seems "normal" now that a man who thinks in his head that he is a woman is allowed to compete in sports with other woman - so that he can win sky high .....
What about women who think in their heads that they are men - do they then compete with men in sports - so they will lose "sky-deep" - if one can say so?

And what about those non-binary people - can they choose with whom they would wish to compete?
Or is there a special non-binary league in football etc?