Showing distinctness of surface links by taking 2-dimensional braids

For an oriented surface link $S$, we can take a satellite construction called a 2-dimensional braid over $S$, which is a surface link in the form of a covering over $S$. We demonstrate that 2-dimensional braids over surface links are useful for showing the distinctness of surface links. We investigate non-trivial examples of surface links with free abelian group of rank two, concluding that their link types are infinitely many.