how i convert string index to integer index

You need to self join your dataframe with a grouped version of itself. This is achievable by using pd.merge() with your dataframe as the left table variable and groupby dataframe as your right table variable.

>>> df
   cater  rating
0      1     2.5
1      2     4.0
2      3     3.0
3      1     4.0
4      2     2.5
5      2     4.0
>>> df2 = df.groupby('cater', as_index=False).mean()
>>> df2
   cater  rating
0      1    3.25
1      2    3.50
2      3    3.00
>>> df3 = pd.merge(df, df2, on='cater', suffixes=('', '_average'))
>>> df3.assign(cater = df3['cater'].map({1:'a',2:'b',3:'c'}))
  cater  rating  rating_average
0     a     2.5            3.25
1     a     4.0            3.25
2     b     4.0            3.50
3     b     2.5            3.50
4     b     4.0            3.50
5     c     3.0            3.00

You can use this,

import pandas as pd

Input = "yourdataframe.xlsx"

df1 = pd.read_excel(Input, sheet_name = ' ')

df2 = df1.groupby('cater', as_index=False).mean().reset_index(drop = True)

df3 = pd.merge(df1, df2, on = 'cater', how = 'left')

df3.columns = ['cater', 'rating', 'average']