She has 12 moves.
So that is 2^12 possible paths
Let her take r moves clockwise (positive) and 12-r moves anti-clockwise. (negative)
So she will end up at r - (12-r) = 2r-12 in a positive directions which is
2r-12 (mod6) = 2r (mod6)
2r(mod6) = 0 when r = 0,3,6,9,12
If r=0 then she goes anticlockwise every time so there is only one way she can do that 1
If r=12 then she goes clockwise every time so there is only one way she can do that 1
If r= 3 there are 12C3 = 220 possible paths
If r = 9 there are also 220 possible paths
If r=6 there are 12C6 = 924 possible paths
924+440+2 = 1366 possible paths where she will end up back at A
Probability that she will end up back at A
\(=\frac{1366}{2^{12}}\\ =\frac{683}{2^{11}}\\ \approx 0.335 \qquad\text{Correct to 3 decimal places}\)
** I am not claiming to be positive that this is completely correct.