Singular or Balkanized Expansion
An initial single homeworld expansion could take multiple forms:
1) global government with single space agancy
2) multiple powers contributing to single space agency
3) multiple powers but single superpower with space agancy
4) multiple superpowers with multiple space agancies.
In the case of multiple factions with their own space agency there would be a risk of conflict. Historically in situations like this the factions involved would often discuss the allocation of territory in advance to prevent conflict.
example: Portugal and Spain dividing the globe outside Europe in 1494
So for example in Traveller terms if there were two superpowers on the home world they might divide space beyond the home solar system into two sectors and colonize them separately.
A singular space agency is a lot simpler but if a campaign setting does have multiple home world factions each factional expansion will follow the same sequence rules as a single expansion.