The abbreviation "LLO" after the Group name stands for "Lady Liverpool's Own", a title bestowed upon the Avonside Troop in 1920 and transferred by Lady Liverpool's permission when the Troop was adopted by St. Matthew's Church in 1922.

What we know of the Lady Liverpool:

1. Lady Liverpool was born on the 23rd of May^[1], 1875 the Hon. Annette Louise Monck, daughter to Henry Power Charles Stanley Monck, 5th Viscount Monck & Lady Edith Caroline Sophia Scott. 
2. She married the 2nd Earl of Liverpool, Arthur William de Brito Savile Foljambe (2nd Earl of Liverpool), on 27 July 1897.
3. In 1913, Lord Liverpool was appointed Governor* of New Zealand (a position which he held until 1920), and so Lady Liverpool came to live in New Zealand for that period.
4. During World War 1, Lady Liverpool became very active in urging the women of New Zealand to do their bit for the war effort, authored a knitting book and helped in the efforts to send thousands of pairs of new socks to troops overseas^[2]
5. In 1920 the Lady Liverpool presented a cup to the Avonside Scout Troop, believed to be for "being the most efficient Troop in New Zealand".  Along with this went the title "Lady Liverpool's Own".  When the Avonside Troop combined with the St. Matthew's Parish Troop in 1922 the cup, Troop flag and the special 'LLO' title were transferred by permission (see attached image).
6. She passed away on 25 May 1948 at age 73.

Photographs:

A photo of the Lady Liverpool & Husband, Lord Liverpool, visiting the Red Cross can be found here. 

One of Lady Liverpool & Husband flanked by military personnel here.

A portrait of the Lady Liverpool (as seen above^[4])

Note: Until permission can be sought to use these photos, they must remain as links.

Translation of the letter permitting the transfer of the LLO title and cup to St. Matthew's Scout Troop (our thanks to Jane Teal at the Anglican Archives in Christchurch for helping with the transcription):

Hartsholme Hall

Lincoln

August 7th 1922

Dear Mr. Clibborn

I am so sorry to have been such a long time in answering your letter.

I am sorry to think that the name of the Troop has had to be altered, as I have very affectionate recollections of Avonside & its Church.

I am very glad however to think that the boys are to retain the title of "Lady Liverpool's Own".  I have often thought of them since I left - But I expect most of the Original Troop have now grown up!

We see quite a lot of the Scouts here - Last year there was a big Rally in the (this?) Park.  At present we have a troop camping out here for a week -  About half of them are cripples.  Tomorrow they are coming over to tea.  The weather has not been good & they have had to take refuge in a barn - owing to heavy rain!

I am sending your warrant of appointment which you were kind enough to send me - i was very glad to hear from you & shall be so glad to do any thing I can at any time for the good of the boys -

Y^rs  Truly

Annette Liverpool

It is probable that the the 'park' in this letter refers to the grounds surrounding Hartsholme Hall (which is no longer standing)^[3], where the Foljambes' lived on their return to England.  This area is now referred to as Hartsholme County Park, and was over 2000 acres in size by the end of the Earl and Lady's ownership.

* This title was changed to 'Governor General of New Zealand' in 1917

Trivia: At the time of writing in 2010, the current Lady Liverpool is Comtesse Marie-Ange de Pierredon, second wife to Edward Foljambe, 5th Earl of Liverpool.

References:

[1] thePeerage.com - http://thepeerage.com/p3881.htm#i38801

[2] NZ History - http://www.nzhistory.net.nz/media/photo/knitting-for-empire

[3] City of Lincoln Council - http://www.lincoln.gov.uk/Info_page_two_pic_2_det.asp?art_id=8565&sec_id...

[4] Timeframes Images - Alexander Turnbull Library, Wellington NZ