Why does SELECT results differ between mysql and sqlite?

I'm re-asking this question in a simplified and expanded manner.

Consider these sql statements:

create table foo (id INT, score INT);

insert into foo values (106, 4);
insert into foo values (107, 3);
insert into foo values (106, 5);
insert into foo values (107, 5);

select T1.id, avg(T1.score) avg1
from foo T1
group by T1.id
having not exists (
    select T2.id, avg(T2.score) avg2
    from foo T2
    group by T2.id
    having avg2 > avg1);

Using sqlite, the select statement returns:

id          avg1      
----------  ----------
106         4.5       
107         4.0       

and mysql returns:

+------+--------+
| id   | avg1   |
+------+--------+
|  106 | 4.5000 |
+------+--------+

As far as I can tell, mysql's results are correct, and sqlite's are incorrect. I tried to cast to real with sqlite as in the following but it returns two records still:

select T1.id, cast(avg(cast(T1.score as real)) as real) avg1
from foo T1
group by T1.id
having not exists (
    select T2.id, cast(avg(cast(T2.score as real)) as real) avg2
    from foo T2
    group by T2.id
    having avg2 > avg1);

Why does sqlite return two records?

Quick update:

I ran the statement against the latest sqlite version (3.7.11) and still get two records.

Another update:

I sent an email to sqlite-users@sqlite.org about the issue.

Myself, I've been playing with VDBE and found something interesting. I split the execution trace of each loop of not exists (one for each avg group).

To have three avg groups, I used the following statements:

create table foo (id VARCHAR(1), score INT);

insert into foo values ('c', 1.5)