The book and also the calculation of your friend is wrong. Correct is 11,75 g.
In your friend calculation it is not considered that potassiumoxide has two potassium atoms.
The 0,25 mol potassium is correct. But 1 mol K correspond to 0.5 mol K2O, what means the calculation must be (39+39+16) g/mol x 0,25 mol/2 = 11,75 g
But this exercise is anyway akademic, because potassiumoxide can not be obtained by burning of potassium. The products are potassium peroxide K2O2 and potassium suboxide KO2.