The presence of ask in the pronunciation key in question is either accounting for accents of both Northern and Southern England, where the word has the same vowel as at and arm, respectively, simply copying a relic from when this shift in Southern England was ongoing, or (most likely) giving the reader freedom to interpret it as either vowel (or a whole other one).
The type of notation seen in the excerpt is one devised for the 1757 dictionary Linguæ Britannicæ Vera Pronunciatio and then popularized by Walker, Worcester, and Webster's dictionaries, used until well into the 20th century (and still by American Heritage). Until the late 19th century, the trap–bath split was still ongoing in Southern England, so the separate symbol for ask, bath, dance, etc. emerged.
Like Northern England, North American accents didn't go through the split, except in some parts of New England. However, what we may now call General American didn't emerge as the prestige accent until after World War II, and actors and announcers before then were taught an accent that mimicked non-rhotic Southern British accents, in which the "ask words" were pronounced with an intermediate sound between [æ] (as in at) and [ɑ] (as in arm), as seen here and here (see Penzl 1940 for more discussion).
EDIT: The very book the excerpt appears to be from in fact gives a fairly decent explanation (American IP needed):
The Italian a, numbered 2, is going through a transition in American speech. In British speech it is fixed as a broad sound; the use of a as in father being much more common in England than in America. Italian a as in last, past, fast, grass in this country is hardly heard west of the Hudson. This is neither a defense nor a condemnation of Western speech as against Eastern. Stage conventions and good manners on the platform still demand a liberal broadening of these sounds. Most students will profit by changing in the direction of using a more open Italian a, rather than by drilling themselves in the habit of keeping it flat and thin.
The use of term for illustrating <ẽ> is indeed odd, but this symbol was used to denote the reduced vowel /ə/ before /r/ (see this answer). So murmur was represented as "mûr′mẽr". For most rhotic North American speakers, these sounds are merged (but not for others—to whom hurry and furry do not rhyme and forward and foreword are not homophonous), so that may be the reason the writer chose an inappropriate example.