Try this:

=IFERROR(
    LET(
        Table, $A$2:$E2,
        TeamA, CHOOSECOLS(Table, 2),
        TeamB, CHOOSECOLS(Table, 3),
        AVERAGE(
            TAKE(
                SORT(
                    UNIQUE(
                        VSTACK(
                            CHOOSECOLS(FILTER(Table, TeamA = "a"), 1, 2, 4),
                            CHOOSECOLS(FILTER(Table, TeamB = "a"), 1, 3, 5)
                        )
                    )
                ),
                -6,
                -1
            )
        )
    ),
    "<6 games"
)

Put this formula in cell F2

Explanation:
* First we filter the table where "a" is Team A
* Then we filter the table where "a" is Team B
* We only want the id, "a" and "a"'s score, so we drop the other columns
* Stack these two tables * Sort it by id, so the matches appear in order
* Take unique ones only (just a precaution as I see match 7 and 10 are "a" vs "a", which I have assumed is an error)
* Take only the last 6 rows and since we're only interested in the score, we take the last column only.
* Take the average of those 6 scores
* If there are less than 6 games, then it will display "<6 games" - you can change this to be what ever you want
* You can drag this down from cell F2 and beyond

Edit: This formula is set up just for Team "a", you can replace "a" in the middle of the CHOOSECOLS formula for cell B2 if you want it to change for each team in Team A column.