Get a list of rows starting from the same value as current row in pandas dataframe

I have a dataframe that I'd like to expand with a new column which would contain/match the list of all ids if they fully contain the row string_value

id  string_value
1   The quick brown fox 
2   The quick brown fox jumps  
3   The quick brown fox jumps over 
4   The quick brown fox jumps over the lazy dog
5   The slow 
6   The slow brown fox 

Desired output

id  string_value                                new_columns
1   The quick brown fox                         [2, 3, 4]
2   The quick brown fox jumps                   [3, 4]
3   The quick brown fox jumps over              [4]
4   The quick brown fox jumps over the lazy dog []
5   The slow                                    [6]
6   The slow brown fox                          []

Thanks

Here's another custom function you can consider. Assuming df is this:

   id                                 string_value
0   1                          The quick brown fox
1   2                    The quick brown fox jumps
2   3               The quick brown fox jumps over
3   4  The quick brown fox jumps over the lazy dog
4   5                                     The slow
5   6                           The slow brown fox

The custom function is

def match_string(string_value):
    idx_list = []
    for idx, strg in list(zip(df['id'], df['string_value'])):
        if strg == string_value:
            continue
        if string_value in strg:
            idx_list.append(idx)
    return idx_list

Then use lambda function:

df['new_columns'] = df['string_value'].apply(lambda x: match_string(x))
print(df)

   id                                 string_value new_columns
0   1                          The quick brown fox   [2, 3, 4]
1   2                    The quick brown fox jumps      [3, 4]
2   3               The quick brown fox jumps over         [4]
3   4  The quick brown fox jumps over the lazy dog          []
4   5                                     The slow         [6]
5   6                           The slow brown fox          []