In makeStyles you should define

  card: {
    width: '360px',
    maxWidth: '100%',
    height: 'fit-content',
    '& .buttonContainer': {
      visibility: 'hidden',
    },
    ':hover': {
      '& .buttonContainer': {
        visibility: 'visible',
      },
    },
  },

and the button container should have the class buttonContainer

Working code: https://stackblitz.com/edit/ff53e2-zpmwug?file=src%2Fexample.tsx