Just as Ishmael and Hagar persecuted Isaac, so natural Israel was the particular persecutor of Christians in the early Church. In fact, that was the main problem of the Galatians, who had trouble either (1) from Jewish Christians who were wobbly and never really became children of faith or (2) from Jews who had never adopted the Christian religion and opposed it as a dangerous heresy.
“If ye be Christ’s, then are ye Abraham’s seed, and heirs according to the promise” (Gal. 3:29). The analogy is to The Christ, Head and body. Therefore, Isaac represents not just the Church but Jesus and the Church. Just as Abraham was Isaac’s father, so God is the Father of The Christ.