There are around 1.5 trillion metric tons of carbon locked in the land permafrost of the Arctic Circle region. To put that into perspective, human fossil fuel emissions are around 34 billion metric tons per year of CO2 (9.3 billion metric tons of carbon), which means that there are about 160 years worth of emissions in carbon locked up in the permafrost of the Arctic region.
(The following argument also applies to the methane hydrates of the Arctic Ocean, but I have decided to focus on the land permafrost. It probably also applies to the Greenland ice sheet.)
It is not currently known how much of the carbon currently locked in permafrost will end up in the atmosphere/ocean system. But let's take just 1/4 of it being released, that's an additional burden of 4 decades worth of current human emissions, and a pulse of emitted CO2 has a long lifetime in the atmosphere/ocean system, centuries to millennia. Given that the experts still don't know how much of the carbon will be released I am not going to get into the 'what fraction' question.
However, before proceeding with my argument, I will observe the following: If the permafrost thaws dry then the emissions will mainly be CO2, but there will be the risk of peat fires at the surface which can burn deep, like moorland fires, that's more CO2. On the other hand, if the permafrost thaws wet, anaerobic metabolisation of the thawed carbon based biomass will entail a significant fraction of methane, CH4. Arguably the latter may be worst, but a lot depends on rates and efficiency, again unknowns.
OK, scene set.
What I am asserting is this:
The total loss of permafrost and release of some fraction of the stored carbon into the atmosphere/ocean system is already a post-bifurcation certainty and is not something any practically attainable human effort can stop.
The loss of permafrost is in large part driven by the regional warming due to the loss of Arctic Sea Ice. In a previous post (almost 10 years ago) I discussed an event in a past geological epoch and relevance to the current situation. I referenced a paper which looked at Arctic regional warming within and without Rapid Ice Loss Events (RILEs) within a climate model, and a figure from that is produced below.