Iterating over a DataFrame, evaluating column values, and setting value to a third column

I have been trying to iterate through a DataFrame or Apply a function, in order to change the content in a specific column of the DataFrame based on 2 other columns also in the DataFrame.

I have a df like:

df = pd.DataFrame({'Age_type' : pd.Series(['Adult','Adult','Child','Child']),
     'Gender' : pd.Series(['Female','Male','Female','Female'])
     })

   Gender  Age_type  Group
0  Female  Adult
1  Male    Adult
2  Female  Child
3  Female  Child

And I want to set a group for each case, with this idea:

if gender == 'Female' and age_type == 'Adult': 
   group = 'Group A'
elif gender == 'Female' and age_type == 'Child':
   group = 'Group B'
elif gender == 'Male' and age_type == 'Adult':
   group = 'Group C'
elif gender == 'Male' and age_type == 'Child':
   group = 'Group D'

I have tried to use .apply(function) because as far as I understand, you should never modify a DataFrame while iterating over it (So this would make a for loop not an option ¿?).

I have tried:

def set_group(data):
    gender = data['Gender']
    age_type = data['Age_type']
    if gender == 'Female' and age_type == 'Adult': 
       data['Group'] = 'Group A'
    elif gender == 'Female' and age_type == 'Child':
       data['Group'] = 'Group B'
    elif gender == 'Male' and age_type == 'Adult':
       data['Group'] = 'Group C'
    elif gender == 'Male' and age_type == 'Child':
       data['Group'] = 'Group D'
    return None

df['Group'].apply(set_group)

but I keep getting errors like: TypeError: string indices must be integers, not str

Any idea on how to iterate over a DataFrame, read some columns, and based on that, set the value for another column?

Thanks!