ItsMyOption: in the idealized world of free-market capitalism, where participants are bound strictly to the rule of law (takings law, conservatorship law, shareholder rights, etc.) the answer is the government does NOT "own" anything they could sell without renegotiating some part of the current agreements or running the risk of futher legal action, as well as potentially incurring some reputational risk with free-market players, and investors - especially all those foreign investors the Trump administration is courting to invest in American markets.
But we all know that this "idealized" world does NOT exist, and in this non-idealized world, the government has a history of having figuratively NO BOUNDS to their actions; however, this world is also NOT without risks. In this world I am sure the government, since Trump's administration has been put in power, has been weighing all the different actions; however, their at least self-professed pro-free market, ownership, pro-growth, pro-shareholder, pro-investment stance is highly likely to extremely limit the actions they will choose to take to options that are favorable to current shareholders.
But the other side of that coin is that there is a financial establishment with both needs and power that almost forces the government to not just relinquish the entire stake back to shareholders. Banks, the Fed, MBS holders, and many other participants cannot just be told to sit down, shut-up, and not react to how uplist and release from c'ship affects their holdings, markets, decision-making, and investments.
It is for these reasons above the government must at least appear to care about the market, those players, and how the solution to this thing occurs which I believe means they will highly likely find a way to "own" something they can sell or use to serve as at least the appearance of watching out for these other players or serve as a backstop for the housing/MBS market.
To this end, I believe this is what the government has Sullivan & Cromwell working on - devising some way to convert the liquidation preference (and/or warrants) into some kind of investment vehicle that allows the government to fulfill the above mentioned cover. I believe the minimum amount of new "takings" the better for both maximizing the value of what they either sell OR, better yet, hold on the books as a backstop where the value of what they hold on the books (the above mentioned converted holdings) is based on the sharply rising market value of the actual mimimally diluted shares outstanding in the market.
There is a plethora of variations from this basic outline, but most of the outliers would seem to create either large or unnecessary political, legal, financial, and reputational risk to the success of the Trump administration's economic agenda.
One can avoid reality, but one cannot avoid the consequences of avoiding reality.
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