This was a tricky question…somewhat similar to the Monty Hall Problem (please see https://alokgoyal1971.com/2013/06/09/puzzle-20-the-monty-hall-problem-very-interesting-must-try/). I got a number of answers and a few of them correct in my opinion. The ones that sent the correct answers include Vishv Vivek Sharma (my nephew in Canada) and Girish Tutakne from Singapore.
I am taking the liberty of copying the answer from Girish:
P(Die is actually a 6 / He reports it as 6) = P(Die is a 6 and he reports it as 6) / P(He reports it as 6).
P(Die is a 6 and he reports it as 6) = 1/6 * 3/4 (3/4 is the probability of his speaking the truth)
Denominator is (5/6×1/4)+(1/6×3/4) = 1/3 (first expression is the probability of it not being a 6 [5/6] and then the person lying [1/6] added to the numerator)
Doing the calculations will yield the answer as 3/8.
Hope you enjoyed the puzzle!