In the paper linked below, S. A. Antonyan gives the following proposition without proof (in fact all results are given without proof). I need a proof of this theorem. If anyone has information on this subject, can you comment?

Propoition 7: A $G$-space is compact if and only if it is simultaneously $G$-pseudocompact (it is equivalent $\beta _{G}X=\beta X$, as mentioned in the article) and $G$-Hewitt.

https://www.mathnet.ru/links/d0ee09e8187cd069c75378b50ea4f597/rm3884_eng.pdf