In the paper ["Hasse principle" for $PSL_2 (\mathbb Z)$ and $PSL_2(\mathbb F)$][1] there's a definition of a **Hasse principle for a group $G$**, but I don't completely get it. Is there a more motivated reformulation of this definition?

Why I am interested: local-global principles are often very interesting in arithmetic geometry, so when I noticed a paper with this title I looked at it to see whether this proves something geometric. 

As said below, one can formulate a problem of computing a group $Sha$ defined by a $g$-module $G$ and a family of subgroups $h_i\in G$, but the actual computation in the paper is for a specific choice of $h_i$, and I can't parse if there is an application of interest. Is it so?

(I suspect this problem arises when you try to prove Hasse principle for equations, like $x^n = a$ but with different Galois groups, see [their first paper][2], though the results there have a mistake, corrected in the next one)

 [1]: https://projecteuclid.org/journals/proceedings-of-the-japan-academy-series-a-mathematical-sciences/volume-74/issue-8/Hasse-principle-for-mathrmPSL_2-leftmathbf-Z-rightandmathrmPSL_2-left-mathbfF_p-/10.3792/pjaa.74.130.full
 [2]: https://projecteuclid.org/journals/proceedings-of-the-japan-academy-series-a-mathematical-sciences/volume-73/issue-7/On-Hasse-principle-for-xn--a/10.3792/pjaa.73.143.full