Did the notion of “Youth Group” exist in the early church?
The concept of "youth group" is relatively recent and parallels the development of classroom type education for children starting in the 1800s.
The Biblical model of instruction is more family oriented and follows the apprenticeship model of instruction over the classroom model.
Deuteronomy 6:6-7 And these words, which I command thee this day,
shall be in thine heart: And thou shalt teach them diligently unto
thy children, and shalt talk of them when thou sittest in thine house,
and when thou walkest by the way, and when thou liest down, and when
thou risest up.
The classroom model of instruction has roots in the Hellenistic academy and could be found at the time of Jesus with the adults who were studying from revered Rabbis such as Gamaliel.
Acts 5:34 Then stood there up one in the council, a Pharisee, named
Gamaliel, a doctor of the law, had in reputation among all the people,
and commanded to put the apostles forth a little space;
However, the limitations of such instruction can be seen
1 Corinthians 8:1 Now as touching things offered unto idols, we know
that we all have knowledge. Knowledge puffeth up, but charity
Acts 5:38 And now I say unto you, Refrain from these men, and let
them alone: for if this counsel or this work be of men, it will come
Gamaliel who was considered one of the brightest and best had followed the classroom path of instruction and when it came to something important such as knowing if Jesus was from God, the best he could come up with was "if".
The relationship between Paul and Timothy reflects more of what the apprentice model of instruction was supposed to be.
2 Timothy 2:1-2 Thou therefore, my son, be strong in the grace that
is in Christ Jesus. And the things that thou hast heard of me among
many witnesses, the same commit thou to faithful men, who shall be
able to teach others also.
The word "son" indicates a familial and relational element that was largely displaced as early as the second century as things like catechetical schools (classroom oriented) became more widely used.
The early church did not so much ignore the instruction of children as left their instruction to their parents. However, the nature of "instruction" was changed and as children were being taken out of the family for schooling starting in the 1800s, the church followed the practice with age segregated instruction and activities.