Can Summoned Monsters have their own initiative?

Question

The spell description for summon monster includes this line:
"It appears where you designate and acts immediately, on your turn."

This means that the summons would act on the summoners initiative count... that turn. What about in subsequent turns?

A Summoner's (the class) monsters last minutes / level, instead of rounds, and thus are more likely to stick around for multiple battles. Would they continue to act on the Summoner's turn, or roll for their own initiative?

Additionally, the "Mighty Summons" mythic feat can grant the "Agile template" to a creature, which gives it "Dual Initiative", which seems to demand that a summoned creature gets its own initiative, independent of the Summoner's, even during the battle in which it was summoned.

Answer 1

While a summoned monster does initially act on the turn of the person summoning, it has its own initiative track. If later it delays its action, or uses a spell or ability that changes its initiative, its initiative changes to a different count than the summoner's (or vice versa).

The rule you're quoting only says that it gets to act on the turn it was summoned, on the same initiative as the summoner, not that their initiative can't change individually later on.

Answer 2

This spell summons an extraplanar creature (typically an outsider, elemental, or magical beast native to another plane). It appears where you designate and acts immediately, on your turn. It attacks your opponents to the best of its ability....

Rules as written, summoned monsters share initiative with their summoner, and act during their turn. The summoner can mix actions with the summoned monster (i.e. summoner moves, then monster moves and attacks, then summoner casts a spell). On subsequent rounds, the summoned monster will act at the same time as their summoner. Changes to the summoner's initiative would then also affect the initiative of the summoned monster.

In subsequent combats, the summoned monster continues to share initiative with its master. If a summoned monster has the Dual Initiative feature, it acts twice in one turn, once on the summoner's turn, and again at the summoner's initiative -20.

This also applies to Summon Nature's Ally, Summon Accuser, etc.