How to check if adjacent row values with common group ID in a column are equal in a pandas dataframe?

I have a dataframe consisting of 7 columns, say each FID is a study object. The objects sharing common values in "FamilyID" are within the same "group". The "YDDW" indicates the names of the study objects.

Now I would like to compare whether two adjacent FID's names (values in "YDDW") are the same along the order of date (values in "Order_Date"). If the two adjacent names are the same, then I would like to assign value "A" to the object with earlier date (defined by "Order_Date") in a new column "Classification"; if the adjacent names are different, then assign "B" in the new column.

Below is the snapshot of the dataframe. The two FIDs 1506, 3388 belonging to the group 006 have the same names in "YDDW", then "A" will be assigned to the row of 3388 in "Classification" column; the two objects 40, 2369 within the group 027 have different names in "YDDW", then "B" will be assigned to the row of 2369 in "Classification" column.

How may I implement these? Thanks in advance!

IIUC and to build upon @Quang Hoang comment, you can try this :

d = {True: "A", False: "B"}
​
df["Classification"] = df["YDDW"].eq(df.groupby("FamilyID")["YDDW"].shift()).map(d)

Output :
​

print(df)

   HighOverID  FamilyID   FID  Order_Date  Order_Year  YDDW  FamilyOrder Classification
0        1506         6  1506  2021-08-25        2021  val1            2              B
1        3388         6  3388  2019-01-14        2019  val1            1              A
2          40        27    40  2023-02-23        2023  val2            2              B
3        2369        27  2369  2020-11-10        2020  val3            1              B
4        1203        55  1203  2021-11-24        2021  val4            2              B
5        3238        55  3238  2019-07-09        2019  val4            1              A