```python
from typing import List

def group_batches(jobs: List[List], capacity: int) -> List[List[str]]:
    if not isinstance(jobs, list):
        raise ValueError("Jobs must be a list.")
    if not all(isinstance(job, list) and len(job) == 4 for job in jobs):
        raise ValueError("Each job must be a four-item list.")
    if not all(isinstance(name, str) and name for job in jobs for name in (job[0], job[1])):
        raise ValueError("Job name and group must be non-empty strings.")
    if not all(isinstance(priority, int) for job in jobs):
        raise ValueError("Priority must be an integer.")
    if not all(isinstance(weight, int) and 1 <= weight <= capacity for job in jobs):
        raise ValueError("Weight must be an integer from 1 to the capacity.")
    if len(jobs) != len(set(job[0] for job in jobs)):
        raise ValueError("Job names must be unique.")
    if not isinstance(capacity, int) or capacity <= 0:
        raise ValueError("Capacity must be a positive integer.")

    batches = []
    groups = {group: [] for name, group, _, _ in jobs}
    job_dict = {name: (group, priority, weight) for name, group, priority, weight in jobs}

    while job_dict:
        eligible_jobs = [(job_dict[name], name) for name, group in job_dict.items() if not groups[group]]
        if not eligible_jobs:
            break

        eligible_jobs.sort(key=lambda x: (-x[0][1], -jobs.index((x[1], *x[0]))))
        batch = []
        current_capacity = capacity

        while eligible_jobs and current_capacity >= eligible_jobs[0][0][2]:
            job_info, name = eligible_jobs.pop(0)
            group, _, weight = job_info
            groups[group].append(name)
            batch.append(name)
            current_capacity -= weight

        batches.append(batch)

    return batches
```