The previous episode of Symphogear AXZ ended with a rather explosive debut for a certain character. In this fifth episode, we see the aftermath of that attack, and Elfnein moves ever closer to completing her research.

Adam’s attack obliterated the entire Kazanari Institute HQ
The Symphogear users survived the attack, but Maria’s team aren’t faring so well after forcibly activating their own Gears. Whilst Maria, Kirika and Shirabe are sent to Tokyo to recover, Hibiki, Tsubasa and Chris discuss their enemies with S.O.N.G.

Genjuro declares their encounter with Adam a total defeat
The alchemists’ Faust Robes came about thanks to the power of the philosopher’s stone, and as such, they are able to render the Ignite module entirely ineffective.

We briefly check in on the alchemists
Adam decides to let Saint-Germain and the other two alchemists deal with the Symphogear users. After obtaining certain information from Tiki, he is far more interested in that.

Kirika transformed
We cut to Kirika and Shirabe fighting Alca-Noise. Kirika’s song is one that I really like, though that tends to be my feelings on the entire Symphogear soundtrack.

Both Kirika and Shirabe force themselves to use their Gears
Kirika and Shirabe are actually participating in a battle simulation, as they are determined to push their sync rates with their Gears up without relying on LiNKER. Their battle comes to an abrupt end, and Hibiki, Chris, Maria and Elfnein rush to them.
After seeing Kirika and Shirabe push themselves like they did, Elfnein says that she only needs one final piece of the puzzle to complete the LiNKER – and the key is Airgetlám: Maria’s Symphogear.
Whilst that is happening, Genjuro visits Kamakura. Things don’t really go too well.

Elfnein intends to merge consciousnesses with Maria
By visiting Maria’s consciousness, Elfnein hopes to discover what part of the brain Airgetlam connects with, and use that to complete the LiNKER research.

Miku will always be there for Hibiki to wipe ice cream from her face or listen to her worries
The other girls worry about Maria and Elfnein. Hibiki gets distracted from her ice cream, but Miku is there for her – and not just when she needs ice cream wiped from her face. Whilst eating their food, the girls happen to see a news story on a big screen about Stefan travelling to Japan to get a prosthetic leg.
Chris knows what she did was the right option, but she clearly regrets it a lot.

Elfnein finds herself in Maria’s memories
Elfnein experiences the same sensations that Maria does in her memories, as they are sharing their consciousnesses. We see a couple of different events from Maria’s past, and the younger Maria is almost never without Serena.

Elfnein stumbles into a memory containing Noise
Should Elfnein perish within Maria’s memories, she would never wake up in the real world. That makes it quite troublesome when Elfnein finds herself being pursued by Noise, but a song soon fills the air.

Maria saves Elfnein
The Maria that appears before Elfnein is not a memory – it is the current Maria’s consciousness and she goes wild in her own mind. Maria breaks through the Noise, but then a blue portal swallows both her and Elfnein.

The alchemists summon a gargantuan Alca-Noise
With a new Alca-Noise appearing, Hibiki and Chris are called into action. Kirika and Shirabe return to HQ. Before Hibiki departs, Miku says to Hibiki that she doesn’t have to win; just don’t lose.

Hibiki departs after hearing Miku’s words
The ending credits roll afterwards, but there is one more scene afterwards. Maria and Elfnein happen to encounter a certain ‘passing hero’ from Maria’s past.
This episode was pretty much Elfnein’s time to shine, and I enjoyed what I saw. She may not be actively participating in combat, but she is working hard to ensure that Maria, Kirika and Shirabe can go back to fight as soon as possible.
We also got a nice couple of moments between Hibiki and Miku, and I will always appreciate those.
I’ve already mentioned this, but I really like Kirika’s song – in fact, I’m a fan of her whole transformation sequence.
It won’t be a surprise to learn that I enjoyed this episode – my feelings on Symphogear as a whole remain pretty consistent on that front.
Elfnein is still adorable and her determination to do what she can for Maria’s Squad is super cool. As you mentioned her sisterly relationship with Maria is also adorable.
Every time HibiMiku get screen time together is a win!