finding the average of a range of rows by column data

In working on this project, I have to get averages of sertain subsets of data in the main data file (my main csv that I'm reading in from has about 500 entries)

df = 
Score   Name    Rating
 2         A    5
 3         A    5
 5         A    4
 4         A    5
 3         B    4
 3         B    4
 5         B    5
 2         C    3
 4         C    3
 4         C    4

I've parsed the set to make sure that all the names are uppercase and that all the scores and ratings are numeric this is what I'm starting off with, but I'm still a beginner and struglling though...

    ds = ds.sort_values(by=['Name'])
    counts = ds["Name"].value_counts()

    names = []
    ratings = []
    avg = []
    for i in range(0, hotel_count(filename)):
        avg[i] = pd.rolling_mean(pd.Series(range(counts[i]), window=2)

I'm looking for output that resembles

Name   AverageScore 
A        3.5
B        3.6
C        2.6
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