June (Junius) is dedicated to Juno, the Patroness of Women. It is unclear why Juno is honored by Romans at this time, since only Juno Moneta has a festival day in June. However, marriages in the last two weeks of June were considered especially blessed by this Goddess of Marriage. For more on Juno: God of the Month: Juno Regina
The main focus of June is the Vestalia from June 7 to 15. The Inner Sanctum of the Temple of Vesta, Goddess of the Hearth, was opened to women. The temple was cleaned, purified, and rededicated. (The rubbish was then into the Tiber River.) In the Roman religion, Vesta is the Perpetual Fire, Who keeps the Pact between the People and the Gods. For more on Vesta: God of the Month: Vesta
I have personal cultus to Hercules, who is considered to be one of the early founders of Rome. Some of the Greek mythology of Heracules was grafted onto Hercules, the Roman God Hero. However, Romans had their own particular myths about Him. For example, the focus of Hercules’ worship, the Ara Maxima (the Greatest Altar) is where He killed Cacus, the monster who terrorized the early Romans.
Two temples of Hercules have dedications this month. Hercules Magno Custodi (the Great Custodian) has one on June 4, and Hercules Musarum (of the Muses) on June 29. The first was vowed on the orders of the Sibylline Books in light of Hannibal’s victories against Rome. The second was where poets and others would come to pay their respects to Hercules and the Muses. For more on Hercules: God of the Month: Hercules
Another God, I have a cultus for is Summanus, the God of the Nocturnal Heavens. He ruled the night as Jupiter ruled the day. His festival day is June 20, when people offered round breads imprinted with wheels to Him. For more on Summanus: God of the Month: Summanus
For more on June in Her many aspects: