The name of the city was Elisavetgrad, not Elizabetgrad. It was a regional center of the Elisavetgrad uezd (county) in the Kherson province. The city was recently renamed from Kirovograd to Kropyvnytskyi. JewishGen hasn't changed the name on the website, so be sure to use "Kirovograd" as the search term.
Archangelskaya/Arkhangelskaya (feminine version of Archangelskiy/Arkhangelskiy) is a based on Arkhangelsk, a town in the western part of the Elisavetgrad uezd, which is now known as Novoarkhanhelsk (New Arch Angel)
Kirovohrad, Ukraine: https://www.jewishgen.org/Communities/community.php?usbgn=-1041993
Novoarkhanhelsk, Ukraine: https://www.jewishgen.org/Communities/community.php?usbgn=-1048149
Ukraine SIG - Yelisavetgrad (Kirovograd): https://www.jewishgen.org/Ukraine/GEO_town.asp?id=26
Ukraine SIG - Novoarkhangelsk: https://www.jewishgen.org/Ukraine/GEO_town.asp?id=39