There is this thing call buybacks in which a company buy it's own shares, which increases the Earnings per Share and is meant to raise the stock price.
Historically management haven't been too good, on average, at deciding where to buy, they usually buy shares back when they are too expensive. Dell used to be a good example of that.
Besides what you propose seems to be a good way to bankrupt a company, let's say management offers to buy at 80$. Well ok, but then a crises comes for whatever the reason and their is a sell off in the stock market.
The company has two options either they buy all the shares until they run out of money or they have to reduce the price at which they are willing to buy, in which case they would set up an alarm and more stockholders would sell, and they would run out of money anyway.
And if there a time of economic bonanza, they would need to raise the price not to sell too cheaply but then, the stock market does that already.
And when the bonanza ends, as it always does, it triggers the sell off, which bankrupts the company.