The sidebar on page 462 of the CRB states the following:
"Using an activity is not the same as using any of its subordinate actions. [...] if you used an action that specified, 'If the next action you use is a Strike,' an activity that includes a Strike wouldn't count, because the next thing you are doing is starting an activity, not using the Strike basic action."
The monster ability Push lists the following as its requirement (page 343 of the Bestiary):
"The monster's last action was a success with a Strike that lists Push in its damage entry."
Now consider the roc's Wing Rebuff reaction as described on page 281:
"The roc makes a wing Strike against the triggering creature. If the roc Pushes the creature, it disrupts the triggering move action."
From my understanding of the CRB text, the second sentence does not make any sense, since it should be impossible for the roc to Push a creature in this situation, the Strike being a subordinate action to the Wing Rebuff reaction.
As another example, consider the last sentence of the kraken's Double Attack feature (involving Grab, which has a similar requirement as Push) on page 214:
"If the kraken subsequently uses the Grab action, it Grabs any number of creatures."
Of course, how these cases should be adjudicated is clear: the roc *can* Push after a Wing Rebuff, and the kraken *can* Grab after a Double Attack. However, the wording of these entries has led me to believe that a similar thing should hold in cases where it is not explicitly mentioned, like the snapping flytrap's Hungry Flurry on page 160 or the remorhaz' Thrash on page 280.
My questions, thus, are the following:
1. Should it be possible, in general, for a monster to use a Grab, Push or Knockdown action (or free action from the Improved versions) after a successful Strike, even if said Strike was a subordinate action?
2. If so, should each Strike from the activity count as a separate trigger for the Improved versions (which are free actions), so that, for example, the snapping flytrap could Grab two creatures at once with its Hungry Flurry?
Note that question 2 is really only relevant for the Improved versions, since in the standard case it is quite clear that only one Grab, Push or Knockdown is possible - the second such action would no longer follow any successful Strike. This also means that, say, the sentence quoted above from the kraken's Double Attack action is not superfluous either way.
After reading large parts of the Bestiary, my intuition would be that the intended answer to both questions is "yes". However, I would be interested in other people's input on this.