Use GroupBy.transform with std for repeating aggregate values, so is possible multiple by p column:
df['STD*p'] = df.groupby('G')['p'].transform('std').mul(df['p'])
print (df)
G month p STD*p
0 G1 1 0.040698 0.011816
1 G1 2 0.225641 0.065509
2 G1 3 0.236948 0.068792
3 G1 4 0.119340 0.034647
4 G1 5 0.779272 0.226243
5 G2 1 0.892169 0.439110
6 G2 2 0.062468 0.030746
7 G2 3 0.936044 0.460705
8 G3 1 0.509213 0.098856
9 G3 2 0.476719 0.092548
10 G3 3 0.407300 0.079071
11 G3 4 0.843261 0.163707
12 G4 1 0.882554 NaN
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