The IRS Website discusses the term "integrated auxiliary of a church"; it refers to a class or organization related to a church but are not such organizations themselves.
"'Integrated Auxiliary of a Church' Defined," Internal Revenue Service, 2023, accessed September 27, 2023
The term integrated auxiliary of a church refers to a class of organizations that are related to a church or convention or association of churches, but are not such organizations themselves. In general, the IRS will treat an organization that meets the following three requirements as an integrated auxiliary of a church. The organization must:
Be described both as an Internal Revenue Code section 501(c)(3) organization and be a public charity under Code section 509(a)(1), (2), or (3),
Be affiliated with a church or convention or association of churches, and
Receive financial support primarily from internal church sources as opposed to public or governmental sources.
Men's and women's organizations, seminaries, mission societies and youth groups that satisfy the first two requirements above are considered integrated auxiliaries whether or not they meet the internal support requirement.