The Book "Possessed" about the Real Case Upon which the Movie The Exorcist was based, indicates that the Possessed Boy was only able to received First Holy Communion on First Saturday after a Rosary dedicated to Our Lady of Fatima. In this same year, the Real Sister Lucy was taking her final vows in the Carmelite Order.
Here are some quotations with regard to this matter from the book, Possessed, Thomas B. Allen (New York: Doubleday, 1993), pp. 148-149.
"To prepare Robbie for his first Communion....Robbie lay still on the bed when Bowdern [the exorcist] gave conditional absolution [the boy had not yet made his first confession]....Bowdern moved close, a particle of the Communion host in his outstretched right hand. One of the priests held the linen cloth called a purificatory at Robbie's chin. Robbie [the pseudonymous name of the possessed boy] became a whirlwind of failing arms and legs. Bowdern pressed closer and placed the particle in Robbie's mouth. He spat it out. A deft move of the purificator and the particle landed on the cloth. Bowdern picked it up and tried again. Once more Robbie spat it out and the purificator caught it. During the next two hours Bowdern tried twice more. Both times Robbie spat out the particle, which was caught on the purificator. O'Flaherty [a parish priest at St. Francis Xavier Church at St. Louis University] remarked that the day was the first Saturday of the month, a day on which services for Our Lady of Fatima were conducted in many churches, including Xavier [Church]. He suggested that they say a rosary in honor of Our Lady of Fatima. When the three priests finished the rosary, Bowdern tried for the fifth time and Robbie swallowed the particle. He had made his first Holy Communion."