So many things were not addressed, so a precise answer is not possible.
But to try to raise the proper questions you should be thinking about:
Should the school have known the permission slip was forged?
Was the forgery particularly bad, and the school was lax in not examining it? Did the student have a history of forging slips that the school should have been aware of?
If the school was negligent in accepting an obviously bad signature, they may find their exposure is increased. If the school had no reasonable way to know the slip was forged, they were acting reasonably in taking the student on an excursion.
Was the injury typical, foreseeable and recoverable?
Such as a broken ankle on a hike? Minor accidents happen even when all reasonable precautions are taken. The injury will heal with time and care.
Was the activity that lead to the injury inherently risky / dangerous?
There is definitely a question of if the school took all reasonable precautions. Even if permission was legitimately given, the school is responsible for taking reasonable precautions, especially if the activity has inherit and obvious dangers. (for example, river-rafting or rock climbing)
What sort of "responsibility" are you interested in?
If you're asking who is financially responsible for the cost of treating the injury, then regardless of how it occurred, it would likely fall to the child's health insurance (presumably provided by the parents).
If the school was truly negligent in allowing a forged permission slip to a dangerous activity, then they could be found responsible for extraordinary costs associated with the injury, other costs (pain, suffering, loss of opportunity, emotional consequences, etc) and perhaps even punitive damages.
If you're suggesting that someone might be criminally responsible, then a very high bar would need to be cleared. It would need to be proven that a school representative (eg. teacher or administrator) deliberately put the kid in danger for some reason, knowing what the likely outcome would be. That standard seems extremely unlikey to be met.