A server push is effectively the response to a request that the client never sent!
Server pushes are only allowed to happen if the client side has agreed to them. In HTTP/3 the client even sets a limit for how many pushes it accepts by informing the server what the max push stream ID is. Going over that limit will cause a connection error.
If the server deems it likely that the client wants an extra resource that it hasn't asked for but ought to have anyway, it can send a
PUSH_PROMISEframe (over the request stream) showing what the request looks like that the push is a response to, and then send that actual response over a new stream.
Even when pushes have been said to be acceptable by the client before-hand, each individual pushed stream can still be canceled at any time if the client deems that suitable. It then sends a
CANCEL_PUSHframe to the server.
Ever since this feature was first discussed in the HTTP/2 development and later after the protocol shipped and has been deployed over the Internet, this feature has been discussed, disliked and pounded up in countless different ways in order to get it to become useful.
Pushing is never "free", since while it saves a half round-trip it still uses bandwidth. It is often hard or impossible for the server-side to actually know with a high level of certainty if a resource should be pushed or not.