async def test_websocket(aiohttp_client, loop):
    app = web.Application()
    app.router.add_get('/', index)
    client = await aiohttp_client(app)
    headers = {'Authorization': 'Token bG912312n3aasdW46cGFzcw=='}
    resp = await client.ws_connect('/', headers=headers)

# в сокете вот так вывожу
pprint(ws_current.headers)
pprint(ws_current.get('Authorization'))

# получаю
"""
<CIMultiDict('Upgrade': 'websocket', 'Connection': 'upgrade', 'Sec-WebSocket-Accept': '+m42KK2222Q9SkaxOLqDCUvu4bo=', 'Content-Type': 'application/octet-stream', 'Date': 'Sat, 22 Feb 2020 07:39:02 GMT', 'Server': 'Python/3.7 aiohttp/3.6.2')>
None
"""